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Jevons, William Stanley 


The principles of science 











Jevons, William Stanley, 1835-1882. 

The principles of science ; a treatise on logic and scien- 
tific method, by W. Stanlej'^ Jevons ... London, Macmil- 
lan and co., limited ; New York, The Macmillan co., 1900. 

xliv, 786 p. incl. front. 19i'-. 

1. Logic. 2. Science — Methodology. i. Title. 

Library of Congress 





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Firtt Edition (2 voU. %vo\ 1874. 

Second Bdition (I vol. erown Svo), 1S77. 

Urorinted wOA eorreetUmM, 1879, 1888, 1887, 1892, 190a 

JUpHnttd (8»o), 1906, 1907, 1918, 1920. 


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It may be truly asserted that the rapid progress of the 
physical sciences during the last three centuries has not 
been accompanied by a corresponding advance in the 
theory of reasoning. Physicists speak familiarly of 
Scientific Method, but they could not readily describe 
what they mean by that expression. Profoundly engaged 
in the study of particular classes of natural phenomena, 
they are usually too much engrossed in the immense and 
ever-accumulating details of their special sciences to 
generalise upon the methods of reasoning which they 
unconsciously employ. Yet few will deny that these 
methods of reasoning ought to be studied, especially by 
those who endeavour to introduce scientific order into less 
successful and methodical branches of knowledge. 

The application of Scientific Method cannot be re- 
stricted to the sphere of lifeless objects. We must sooner 
or later have strict sciences of those mental and social 
phenomena, which, if comparison be possible, are of more 
interest to us than purely material phenomena. But it 
is the proper course of reasoning to proceed from the 
known to the unknown — from the evident to the obscure 
—from the material and palpable to the subtle and 
refined. The physical sciences may therefore be properly 

« I 


1 1 






made the practice-ground of tlie reasoning powers, because 
they furnish us with a great body of precise and successful 
investigations. In these sciences we meet with happy 
instances of unquestionable deductive reasoning, of ex- 
tensive generalisation, of happy prediction, of satisfactory 
verification, of nice calculation of probabilities. We can 
note how the slightest analogical clue has been followed 
up to a glorious discovery, how a rash generalisation has 
at length been exposed, or a conclusive cxpcrimcntum 
crucis has decided the long-continued strife between two 
rival theories. 

In following out my design of detecting the general 
methods of inductive investigation, I have found that the 
more elaborate and interesting processes of quantitative 
induction have their necessary foundation in the simpler 
/ science of Formal Logic. The earlier, and probably by^ 
Mar the least attractive part of this work, consists, there- 
fore, in a statement of the so-called Fundamental Laws 
of Thought, and of the all-important Principle of Substi- 
tution, of which, as I think, all reasoning is a develop- 
ment. The whole procedure of inductive inquiry, in its 
most complex cases, is foreshadowed in the combinational 
view of Logic, which arises directly from these fundamental 
principles. Incidentally I have described the mechanical 
arrangements by which the use of the iraportant form 
called the Logical Alphabet, and the whole working of 
the combinational system of Formal Logic, may be ren- 
dered evident to the eye, and easy to the mind and 

The study both of Formal Logic and of the Theory of 
Probabilities has led me to adopt the opinion that there 
is no such thing as a distinct method of induction as 
contrasted with deduction, but that induction is simply 
an inverse employment of deduction. Within the last 
century a reaction has been setting in against the purely 
empirical procedure of Francis Bacon, and physicists have 





learnt to advocate the use of hypotheses. I take the 
extreme view of holding that Francis Bacon, although he 
correctly insisted upon constant reference to experience, 
had no correct notions as to the logical method by which 
from particular facts we educe laws of nature. I endea- 
vour to show that hypothetical anticipation of nature is 
an essential part of inductive inquiry, and that it is the 
Newtonian method of deductive reasoning combined with 
elaborate experimental verification, which has led to all 
the great triumphs of scientific research. 

In attempting to give an explanation of this view of 
Scientific Method, I have first to show that the sciences 
of number and quantity repose upon and spring from the 
simpler and more general science of Logic. The Theory 
of Probability, which enables us to estimate and calculate 
quantities of knowledge, is then described, and especial 
1 attention is drawn to the Inverse Method of Probabilities, 
\ which involves, as I conceive, the true principle of in- 
/ ductive procedure. No inductive conclusions are more 
than probable, and I adopt the opinion that the theory of 
probability is an essential part of logical method, so that 
the logical value of every inductive result must be deter- 
mined consciously or unconsciously, according to the 
principles of the inverse method of probability. 

The phenomena of nature are commonly manifested 
in quantities of time, space, force, energy, &c., and the 
observation, measurement, and analysis of the various 
quantitative conditions or results involved, even in a 
simple experiment, demand much employment of system- 
atic procedure. I devote a book, therefore, to a simple 
and general description of the devices by which exact 
measurement is effected, errors eliminated, a probable 
mean result attained, and the probable error of that mean 
ascertained. I then proceed to the principal, and probably 
the most interesting, subject of the book, illustrating 
successively the conditions and precautions requisite for 



i 7> 

•J I 





\ ' 


accurate observation, for successful experiment, and for 

the sure detection of the quantitative laws of nature. 

As it is impossible to comprehend aright the value of 

quantitative laws without constantly bearing in mind the 

\ degree of quantitative approximation to the truth probably 

s attained, I have devoted a special chapter to tlie Theory 

S)f Approximation, and however imperfectly I may have 

treated this subject, I must look upon it as a very essential 

Dart of a work on Scientific Method. 

It then remains to illustrate the sound use of hypo- 
thesis, to distinguish between the portions of knowledge 
wliich we owe to empirical observation, to accidental dis- 
covery, or to scientific prediction. Interesting questions 
arise concerning the accordance of quantitative theories 
and experiments, and I point out how the successive 
'verification of an hypothesis by distinct methods of ex- 
periment yields conclusions approximating to but never 
attaining certainty. Additional illustrations of the geneml 
procedure of inductive investigations are given in a 
chapter on the Character of the Experimentalist, in which 
I endeavour to show, moreover, that the inverse use of 
deduction was really the logical method of such gi-eat 
masters of experimental inquiry as Newton, Huyghens, 
and Faraday. 

In treating Generalisation and Analogy, I consider the 
precautions requisite in inferring from one case to another, 
/ or from one part of the universe to another part ; the 
( validity of all such inferences resting ultimately upon 
\^he inverse method of probabilities. The treatment of 
Exceptional Phenomena appeared to afford an interesting 
subject for a further chapter illustrating the various modes 
in which an outstanding fact may eventually be explained. 
The formal part of the book closes with the subject of 
Classification, which is, however, very inadequately treated. 
I have, in fact, almost restricted myself to showing that 
all classification is fundamentally carried out upon the 



principles of Formal Logic and the Logical Alphabet 
described at the outset. 

In certain concluding remarks I have expressed the 
conviction which the study of Logic has by degrees 
forced upon my mind, that serious misconceptions are 
enteiiained by some scientific men as to the logical value 
of our knowledge of nature. We have heard much of 
what has been aptly called the Eeign of Law, and the 
necessity and uniformity of natural forces has been not 
uncommonly interpreted as involving the non-existence 
of an intelligent and benevolent Power, capable of inter- 
fering with th'iB course of natural events. Fears have 
been expressed that the progress of Scientific Method 
must therefore result in dissipating the fondest beliefs 
of the human heart. Even the 'Utility of lleligion' is 
seriously proposed as a subject of discussion. It seemed 
to be not out of place in a work on Scientific Method to 
allude to the ultimate results and limits of that method. 
I fear that I have very imperfectly succeeded in expressing 
my strong conviction that before a rigorous logical scrutiny 
the Keign of Law will prove to be an unverified hypo- 
thesis, the Uniformity of Nature an ambiguous expression, 
the certainty of our scientific inferences to a great extent 
a delusion. The value of science is of course very high, 
while the conclusions are kept well within the limits of 
the data on which they are founded, but it is pointed out 
that our experience is of the most limited character com- 
pared with what there is to learn, while our mental powers 
seem to fall infinitely short of the task of comprehending 
and explaining fully the nature of any one object. I 
draw the conclusion that we must interpret the results/ 
of Scientific Method in an afifirmative sense only. Ours] 
must be a truly positive philosophy, not that false nega-/ 
tive philosophy which, building on a few material facts,^ 
presumes to assert that it has compassed the bounds 
of existence, while it nevertheless ignores the most/ 





unquestionable phenomena of tlie human mind and feel- 


It is approximately certain that in freely employing 
illustrations drawn from many different sciences, I have 
frequently fallen into errors of detail. In this respect I 
must throw myself upon the indulgence of the reader, 
who will bear in mind, as I hope, that the scientific fact^ 
are generally mentioned purely for the purpose of illus- 
tration, so that inaccuracies of detail will not in the 
majority -^f cases affect the truth of the general principles 





December i«;, i87». 

M 1 




Few alterations of importance have been made in pre- 
paring this second edition. Nevertheless, advantage has 
bcxjn taken of the opportunity to revise very carefully 
both the language and the matter of the book. Cor- 
respondents and critics having pointed cut inaccuracies 
of more or leis importance in the first edition, suitable 
corrections and emendations have been made. I am under 
obligations to Mr. C. J. Monro, M.A., of Bamet, and to 
Mr. W. H. Brewer, MA., one of Her Majesty's Inspectors 
of Schools, for numerous corrections. 

Among several additions which have been made to the 
text, I may mention the abstract (p. 143) of Professor 
Clifford's remarkable investigation into the number of 
types of compound statement involving four classes of 
objects. This inquiry carries forward the inverse logical 
problem described in the preceding sections. Again, the 
need of some better logical method than the old Barbara 
Celarent, &c., is strikingly shown by Mr. Venn's logical 
problem, described at p. 90. A great number of candidates 
in logic and philosophy were tested by Mr. Venn with this 
problem, which, though simple in reality, was solved by 
very few of those who were ignorant of Boole's Logia 
Other evidence could be adduced by Mr. Venn of the need 
for some better means of logical training. To enable the 







logical student to test his skUl in the solution of inductive 
logical problems, I have given (p. 127) a series of ten 
problems graduated in difficulty. 

To prevent misapprehension, it should be mentioned 
that, throughout this edition, I have substituted the name 
Zoffical Alphabet for Zor/ical Ahecedarium, the name applied 
in the first edition to the exhaustive series of logical 
combinations represented in terms of A, B, C, D (p. 94). 
It was objected by some readers that Ahecedarium \a a 
bug and unfamiliar name. 

To the chapter on Units and Standards of Measure- 
ment, I have added two sections, one (p. 325) containing 
a bnef statement of the Theory of Dimensions, and the 
other (p. 319) discussing Professor Clerk Max weU's very 
original suggestion of a Natural System of Standards for 
the measurement of space and time, depending upon the 
length and rapidity of waves of light. 

In my description of the Logical Machine in tho 
Philosophical Trarisactions (vol. 160, p. 498), I said— 
" It is rarely indeed that any invention is made without 
some anticipation being sooner or later discovered ; but up 
to the present time I am totally unaware of even a single 
previous attempt to devise or construct a macliiue which 
should perform the operations of logical inference ; and it 
IS only, I believe, in the satirical writings of Swift that an 
allusion to an actual reasoning machine is to be found." 
Before the paper was printed, however, I was able to refer 
(p. 518) to the ingenious designs of the late Mr. Alfred 
Smee as attempts to represent thought mechanically. 
Mr. Smee's machines indeed were never constructed, and, 
if constructed, would not have performed actual logical 
inference. It has now just come to light, however, that 
the celebrated Lord Stanhope actually did construct a 
mechanical device, capable of representing syUogistic 
mferences in a concrete form. It appears that logic was 
one of the favourite studies of this truly original and 
ingenious noblecmu. There remain fragments of a logical" 





work, printed by the Earl at his own press, which show 
that he had arrived, before the year 1800, at the principle 
of the quantified predicate. He puts forward this prin- 
ciple in the most explicit manner, and proposes to employ 
it throughout his syllogistic system. Moreover, he con- 
verts negative propositions into affirmative ones, and 
represents these by means of the copula " is identic with." 
Thus he anticipated, probably by the force of his own 
unaided insight, the main points of the logical method 
originated in the works of George Bentham and George 
Boole, and developed in this work. Stanhope, indeed, has 
no claim to priority of discovery, because he seems never 
to liave published his logical writings, although they were 
put into print. There is no trace of them in the British 
Museum Library, nor in any other library or logical work, 
80 far as I am aware. Both the papers and the logical 
contrivance have been placed by the present Earl Stanhope 
in the hands of the Kev. Eobert Harley, F.R.S., who will, 
I hope, soon publish a description of them.^ 

By the kindness of Mr. Harley, I have been able to 
examine Stanhope's logical contrivance, called by him the 
Demonstrator. It consists of a square piece of bay- wood 
with a square depression in the centre, across which two 
slides can be pushed, one being a piece of red glass, and 
the other consisting of wood coloured gray. The extent 
to which each of these slides is pushed in is indicated by 
scales and figures along the edges of the aperture, and the 
simple rule of inference adopted by Stanhope is : " To the 
gi-ay add the red and subtract the holon^' meaning by 
holon ipkov) the whole width of the aperture. This rule 
of inference is a curious anticipation of De Morgan's 
numerically definite syllogism (see below, p. 168), and of 
inferences founded on what Hamilton called " Ultra-total 
distribution." Another curious point about Stanhope's 

* Since tho above was written Mr. Harley has read an account of Stan- 
hope's logical remains at the Dublin Meeting (1878) of the British 
Association. The mper will be printed in Mind. (Note added November. 
l87».) 1 r- f 







device is, that one slide can be drawu out and pushed in 
again at right angles to the other, and the overlappino 
part of the slides then represents the probability of a 
conclusion, derived from two premises of which the pro- 
babilities are respectively represented by the projecting 
parte of the slides. Thus it appeal^ that Stanhope had 
studied the logic of probabHity as well as that of certainty 
here agam anticipating, however obscurely, the recent 
progress of logical science. It wiU be seen, however, that 
between Stanhope's Demonstrator and my Logical Machine 
there is no resemblance beyond the fact that they both 
perform logical inference. 

In the first edition I inserted a section (vol i p. 25), on 
Anticipations of the Principle of Substitution," and I 
have reprmted that section unchanged in this edition 
(p. 21). I remark therein that, " In such a subject as lo-ic 
It IS hardly possible to put forth any opinions which ha've 
not been m some degree previously entertained. The 
germ at least of every doctrine wiU be found in eariier 
wntmgs, and novelty must arise chiefly in the mode of 
harmonising and developing ideas." I point out as 
Professor T. M Lindsay had previously done, that Beneke 
had employed the name and principle of substitution, and 
that doctrines closely approximating to substitution were 
stoted by the Port Eoyal Ix>gicians more than 200 years 

I have not been at aU surprised to learn, however, that 
other logicians have more or less distinctly stated this 
principle of substitution during the last two centuries 
As my friend and successor at Owens CoUege, Professor 
Adamson, has discovered, this principle can be traced back 
to no less a philosopher than Leibnitz. 

The remarkable tract of Leibnitz,i entitled "Non inelegans 
Specmien Demonstrandi in Abstractis," commences at once 
with a defimtion corresponding to the principle :— 

i^tT "^^ ^^'"^"^ ^ ^^. Krdxuaan. Par. I. Ben,liBi, 



" Eadem sunt quorum unum potest substitui alteri salva 
veritate. Si sint A et B, et A ingrediatur aliquam pro-" 
positionem veram, et ibi in aliqiio loco ipsius A pro ipso 
substituendo B fiat nova propositio seque itidem vera, idque 
semper succedat in quacunque tali propositione, A et B 
dicuntur esse eadem ; et contra, si eadem sint A et B, 
procedet substifutio quam dixi." 

Leibnitz, then, explicitly adopts the principle of sub- 
stitution, but he puts it in the form of a definition, saying 
that those things are the same which can be substituted 
one for the other, without affecting the truth of the 
proposition. It is only after having thus tested the same- 
ness of things that we can turn round and say that A and 
B, being the same, may be substituted one for the other. 
It would seem as if we were here in a vicious circle ; for 
we are not aUowed to substitute A for B, unless we have 
ascertained by trial tliat the result is a true proposition. 
Tlie difficulty does not seem to be removed by Leibnitz' 
proviso, "idque semper succedat in quacunque tali pro- 
positione." How can we learn that because A and B may 
be mutually substituted in some propositions, they may 
therefore be substituted in others ; and what is the criterion 
of likeness of propositions expressed in the word " tali " ? 
Whether the principle of substitution is to be regarded as a 
postulate, an axiom, or a definition, is just one of tliose fun- 
damental questions which it seems impossible to settle in the 
present position of philosophy, but this uncertainty will not 
prevent our making a considerable step in logical science. 

Leibnitz proceeds to establish in the form of a theorem 
what is usually taken as an axiom, thus (Opera, p. 95) : 
•• Theorema 1. Quae sunt eadem uni tertio, eadem sunt 
inter se. Si A cc B et B ex: C, erit A ex C, Nam si in 
pi-opositione A cc B (vera ea hypothesi) substituitur C in 
locum B (quod facerc licet per Def. I. quia B oc C ex 
hypothesi) fiet A cc C. Q. E. Dcm." Thus Leibnitz 
precisely anticipates the mode of treating inference with 
two simple identities described at p. 5 1 of this work. 













Even the mathematical axiom that 'equals added to 
equals make equals/ is deduced from the principle of 
substitution. At p. 95 of Erdmann's edition, we find : " Si 
eideni addantur coincidentia fiunt coincidentia. SiAoiB, 
erit A + C oz B ■\- 0. Nam si in propositione A ■{■ C cc A 
-f C (quae est vera per se) pro A semel substituas /? (quod 
facere licet per Def. I. quia A (x B) ^et A -\- G o: B •{ C 
Q. K Dem." This is unquestionably the mode of deducing 
the several axioms of mathematical reasoning from the 
higher axiom of substitution, which is explained in the 
section on mathematical inference (p. 162) in this work, 
and which had been previously stated in my StcbstittUion 
of Similars, p. 16. 

Tliere are one or two other brief tracts in which Leibnitz 
anticipates the modern views of logic Thus in the 
eighteenth tract in Erdmann's edition (p. 92), called 
"Fundamenta Calculi Ratiocinatoris, he says: "Inter ea 
quorum unum alteri substitui potest, sal vis calculi legibus, 
dicetur esse gequipollentiam." There is evidence, also, that 
he had arrived at the quantification of the predicate, and 
that he fully understood the reduction of the universal 
affirmative proposition to the form of an equation, which is 
the key to an improved view of logic. Thus, in the tract 
entitled "Difficultates Qujedam Logicae,"* he says : "Omne-<4 
est ^; id est equivalent AB et A, sen A non B est non-ens." 

It is curious to find, too, that Leibnitz was fully ac- 
quainted with the Laws of Commutativeness and " Simpli- 
city " (as I have called the second law) attaching to logical 
symbols. In the * Addenda ad Specimen Calculi Univer- 
salis" we read as follows.* " Transpositio literarum in 
eodem termino nihil mutat, ut ah coincidet cum ha, sen 
animal rationale et rationale animal." 

** Repetitio ejusdem litene in eodem termino est inutilis, 
ut h est aa; vel hh est a; homo est animal animal, vel 
homo homo est animaL Sufficit cnim dici a est h, seu 
homo est animal." 

Comparing this with what is stated in Boole's Mathe- 
matical Analysis of Logic, pp. 17-18, in his Laws of 
Thov^ht, p. 29, or in this work, pp. 32-35, we find that 
Leibnitz had arrived two centuries ago at a clear perception 
of the bases of logical notation. When Boole pointed out 
that, in logic, axe = a?, this seemed to mathematicians to be 
a paradox, or in any case a wholly new discovery; but 
here we have it plainly stated by Leibnitz. 

Tlie reader must not assume, however, that because 
Leibnitz correctly apprehended the fundamental principles 
of logic, he left nothing for modern logicians to do. On 
the contrary, Leibnitz obtained no useful results from his 
definition of substitution. When he proceeds to explain 
the syllogism, as in the paper on " Definitiones Logicae," ^ 
he gives up substitution altogether, and falls back upon 
the notion of inclusion of class in class, saying, " Inclu- 
dens includentis est includens inclusi, seu si A includit B 
ct B includit G, etiam A includet G." He proceeds to 
make out certain rules of the syllogism involving the 
distinction of subject and predicate, and in no important 
respect better than the old rules of the syllogism. 
Leibnitz* logical tracts are, in fact, little more than brief 
memoranda of investigations which seem never to have 
been followed out They remain as evidence of his 
wonderful sagacity, but it would be difficult to show that 
they have had any influence on the progress of logical 
science in recent times. 

I should like to explain how it happened that these 
logical writings of Leibnitz were unknown to me, until 
within the last twelve months. I am so slow a reader 
of Latin books, indeed, that my overlooking a few pages 
of Leibnitz' works would not have been in any case 
surprising. But the fact is that the copy of Leibnitz' 
works of which I made occasional use, was one of the 
edition of Dutens, contained in Owens College Library. 
The logical ti-acts in question were not printed in that 

* Erdmann, p. 102. 

• Ibid p. 98. 

* £rdjnann, p. loa 

b 2 










edition, and with one exception, they remained in manu- 
script in the Eoyal Library at Hanover, until edited by 
Erdmann, in 1839-40. The tract " DifiBcultates Queedam 
Logicse," though not known to Dutens, was published by 
Kaspe in 1765, in his collection called (Euvres PhUo- 
sophiques tie feu M^' Leibnitz: but this work had not 
come to my notice, nor does the tract in question seem 
to contain any explicit statement of the principle of 

It is, I presume, the comparatively recent publication of 
Leibnitz' most remarkable logical tracts which explains 
the apparent ignorance of logicians as regards their con- 
tents and importance. The most learned logicians, such 
as Hamilton and Ueberweg, ignore Leibnitz* principle of 
substitution. In the Appendix to the fourth volume of 
Hamilton's Lectures on Meta^physics and Logic, is given 
an elaborate compendium of the views of logical writers 
concerning the ultimate basis of deductive reasoning. 
Leibnitz is briefly noticed on p. 319, but without any 
hint of substitution. He is here quoted as saying, " What 
are the same with the same third, are the same with each 
other ; that is, if ^ be the same with B^ and G be the 
same with B, it is necessary that A and C should also 
be the same with one another. For this principle flows 
immediately from the principle of contradiction, and is 
the ground and basis of all logic ; if that fail, there is no 
longer any way of reasoning with certainty." This view 
of the matter seems to be inconsistent with that which he 
adopted in his posthumous tract. 

Dr. Thomson, indeed, was acquainted with Leibnitz* 
tracts, and refers to them in his Outline of the Necessary 
Laws of Thought. He calls them valuable ; nevertheless, 
he seems to have missed the really valuable point ; for in 
making two brief quotations,^ he omits all mention of the 
principle of substitution. 

Ueberweg is probably considered the best authority 

> Fifth Edition, i860, p. 158. 

concerning the history of logic, and in his well-known 
System of Logic and History of Logical Doctrines^ he gives 
some account of the principle of substitution, especially 
as it is implicitly stated in the Port Eoyal Logic. But he 
omits all reference to Leibnitz in this connection, nor does 
he elsewhere, so far as I can tind, supply the omission. 
His English editor. Professor T. M. Lindsay, in referring to 
my Svhstitution of Similars, points out how I was antici- 
pated by Beneke ; but he also ignores Leibnitz. It is thus 
apparent that the most learned logicians, even when writing 
especially on the history of logic, displayed ignorance of 
Leibnitz' most valuable logical writings. 

It has been recently pointed out to me, however, that 
the Rev. Robert Harley did draw attention, at the Not- 
tingham Meeting of the Biitish Association, in 1866, to 
Leibnitz' anticipations of Boole's laws of logical notation,* 
and I am informed that Boole, about a year after the pub- 
lication of his Laws of Thought, was made acquainted with 
these anticipations by R. Leslie Ellis. 

There seems to have been at least one other German 
logician who discovered, or adopted, the principle of sub- 
stitution. Reusch, in his Systema Logicum, published in 
1734, laboured to give a broader basis to the Dictum de 
Omni et Nullo. He argues, that " the whole business of 
ordinary reasoning is accomplished by the substitution of 
ideas in place of the subject or predicate of the funda- 
mental proposition. This some call the equation of thoughts." 
But, in the hands of Reusch, substitution does not seem to 
lead to simplicity, since it has to be carried on according 
to the rules of Equipollence, Reciprocation, Subordination, 
and Co-ordination.' Reusch is elsewhere spoken of * as the 
" celebrated Reusch " ; nevertheless, I have not been able to 

* Section 120. 

« See his "Remarks on Boole's Mathematical Analysis of Loric." 
fP^t of tM s^h Meeting of the British Association, Transactuyns of ik(. 
Sections, pp. 3—6. / "^ 

' Hamilton's Lectures, vol. iv. p. ug. 

* IbicL p. 326. 








find a copy of his book in London, even in the British 
Museum Library; it is not mentioned in the printed 
catalogue of the Bodleian Libraiy; Messrs. Asher have 
failed to obtain it for me by advertisement in Germany ; 
and Professor Adamson has been equally unsuccessful. 
From the way in which the principle of substitution is 
mentioned by Keusch, it would seem likely that other 
logicians of the early part of the eighteenth century were 
acquainted with it ; but, if so, it is still more curious that 
recent historians of logical science have overlooked the 

It is a strange and discouraging fact, that tnie views of 
logic should have been discovered and discussed from one 
to two centuries ago, and yet should have remained, like 
George Bentham's work in this century, without influ- 
ence.on the subsequent progress of the science. It may 
be regarded as certain that none of the discoverers of 
the quantification of the predicate, Bentham, Hamilton, 
Thomson, De Morgan, and Boole, were in any way assisted 
by the hints of the principle contained in previous writers. 
As to my own views of logic, they were originally moulded 
by a careful study of Boole's works, as fully stated in my 
first logical essay.^ As to the process of substitution, it 
was not learnt from any work on logic, but is simply the 
process of substitution perfectly familiar to mathematicians, 
and with which I necessarily became familiar in the course 
of my long-continued study of mathematics under the late 
Professor De Morgan. 

I find that the Theory of Number, which I explained in 
the eighth chapter of this work, is also partially anticipated 
in a single scholium of Leibnitz. He first gives as an 
axiom the now well-known law of Boole, as follows :— 

" Axioma L Si idem secum ipso sumatur, nihil consti- 
tuitur novum, sen ^ + ^ oc A." Then follows thia 

» Pure TA>gic or t)u Logic of QualUtj apart from Quantity; with 
Remarks onBooU'e System, and on the Helatian of Logic and AlalhmuUice 
London, 1864, p. 3- 

remarkable scholium : " Equidem iu numeris 4 + 4 facit 
8, seu bini nummi binis additi faciunt quatuor nummos, 
sed tunc bini additi sunt alii a prioribus ; si iidem essent 
nihil novi prodiret et perinde esset ac si joco ex tribus 
ovis facere vellemus sex numerando, primum 3 ova, deinde 
uno sublato residua 2, ac denique uno rursus sublato 

Translated this would read as follows : — 

"Axiom I. If the same thing is taken together with 
itself, nothing new arises, or A -h A== A. 

" Scholium. In numbers, indeed, 4+4 makes 8, or two 
coins added to two coins make four coins, but then the 
two added are different from the former ones ; if they were 
the same nothing new would be produced, and it would 
be just as if we tried in joke to make six eggs out of three, 
by counting firstly the three eggs, then, one being removed, 
counting the remaining two, and lastly, one being again 
removed, counting the remaining egg." 

Compare the above with pp. 156 to 162 of the present 

M. Littrd has quite recently pointed out ^ what he thinks 
is an analogy between the system of formal logic, stated 
in the following pages, and the logical devices of the 
celebrated Itaymond Lully. Lully's method of invention 
was described in a great number of mediaeval books, but 
is best stated in hisArs Compendiosa Inveniendi Veritatem, 
seu Ars Magna et Major. This method consisted in placing 
various names of things in the sectors of concentric 
circles, so that when the circles were turned, every possible 
combination of the things was easily produced by mechani- 
cal means. It might, perhaps, be possible to discover in 
this method a vague and rude anticipation of combinational 
logic; but it is well known that the results of Lully's 
method were usually of a fanciful, if not absurd character. 

A much closer analogue of the Logical Alphabet is 
probably to be found in the Logical Square, invented by 

» La Philcsovhic Positive Mai-Juin, 1877, torn, xviii. p. 456. 








} • 

John Christian I^nge, and described in a rare and un- 
noticed work by liira which I have recently fonnd in the 
British Museum.i This squai-e involved the principle of 
bifurcate classification, and was an improved form of the 
Ramean and Porphyrian tree (see below, p. 702). Lange 
seems, indeed, to have worked out his Logical Square 
into a mechanical form, and he suggests that it might l>e 
employed somewhat in the manner of Napier's Rones 
(p. 65). There is much analogy between his Square and 
my Abacus, but Lange had not arrived at a logical system 
enabling him to use his invention for logical inference in 
the maimer of the Logical Abacus. Another work of 
Lange is said to contain the first publication of the well 
known Eulerian diagrams of proposition and syllogism.' 

Since the first edition was published, an important 
work by Mr. George Lewes has appeared, namely, his 
Problems 0/ Life cnui Mind, which to a great extent treats 
of scientific method, and formulates the rules of philo- 
sophising. I should have liked to discuss the bearing 
of Mr. Lewes's views upon those here propounded, but 
I have felt it to be impossible in a book already filling 
nearly 800 pages, to enter upon the discussion of a 
yet more extensive book. For the same reason I have 
not been able to compare my own treatment of the subject 
of probability with the views expressed by Mr. Venn in 
his Logic of Chance. With Mr. J. J. Murphy's profound 
and remarkable works on Hahii and Intelligence, and on 
The Scientific Basis of Faith, I was unfortunately unac- ' 
quainted when I wrote the following pages. They can- 
not safely be overlooked by any one who wishes to 
comprehend the tendency of philosophy and scientific 
method in the present day. 

It seems desirable that I should endeavour to answer 
some of the critics who have pointed out what they 

Svl^*^'*'"'^ ^'^^"^ (?wa<fra« Logiei, &c., Gisste Haasorum, 1714, 
' Sei Ueherxoeg'8 SysUm of Lo^e, &c., translated by Lindsay, p. 302. 

consider defects in the doctrines of this book, especially in 
tlie first part, which treats of deduction. Some of the 
notices of the work were indeed rather statements of its 
contents than critiques. Thus, I am much indebted to 
M. Louis Liard, Professor of Philosophy at Bordeaux, for 
the very careful exposition ^ of the substitutional view of 
logic which he gave in the excellent Revue Philosophique, 
edited by M. Ribot. (Mars, 1877, tom. iii. p. 277.) An 
equally careful account of the system was given by 
M. Riehl, Professor of Philos(^hy at Graz, in his article on 
"Die Englische Logik der Gegenwart," published in the 
Vierteljahrsschrift filr toissenschaftliche Philosophie. ( i Heft, 
Leipzig, 1876.) T should like to acknowledge also the 
careful and able manner in which my book was reviewed 
by the New York Daily Tribune and the New York Times. 
The most serious objections which have been brought 
against my treatment of logic have regard to my failure 
to enter into an analysis of the ultimate nature and origin 
of the Laws of Thought. The Spectator^ for instance, in 
the course of a careful review, says of the principle of 
substitution, " Surely it is a great omission not to discuss 
whence we get this great principle itself; whether it is a 
pure law of the mind, or only an approximate lesson of 
experience ; and if a pure product of the mind, whether 
there are any other products of the same kind, furnished 
by our knowing faculty itself." Professor Robertson, in 
his very acute review,^ likewise objects to the want of 

» Since the above wa.s written M. Liard has republislied this exposition 
M one chanter of an interesting and admirably lucid account of the 
progress of logical science m England. After a brief but clear introduc- 

IndA.t^'I i"^*'^ic%^T,'^^^l''^^'^^^ ^^"' ^°d others concerning 
Sirf « .V^""' w' ^'.f ^ describes m succession the logical systems of 

t^^r.^^r'^'^'l' "^?"^'«'V ^^ J^'"'-^"' ^^^1«' ^"'i that contained in 
the present work. The title of the book is as follows -.-Les Logideiui 

k^i".^ifT ^^r*""^:, »"^^^"^' ^^""^' Professeur de Philos^hie \, 
io7». (A ote added November, 1878.) 

-^xsctotor, September 19 1874, p. 1178. A second portion of the 
review api^eared m the same journal fur September 26, 1874, p. ,204 







psychological and jihilosophical analysis. ** If the book 
really corresponded to its title, Mr. Jevons could hardly 
have passed so lightly over the question, which he does 
not omit to raise, concerning those undoubted principles 
of knowledge commonly called the Laws of Thouglit .... 
Everywhere, indeed, he appears least at ease when he 
touches on questions properly pliilosophical ; nor is he 
satisfactory in his psychological references, as on pp. 4, 5, 
where he cannot commit himself to a statement without 
an accompaniment of 'probably,' 'almost,' or 'hardly.' 
Reservations are often very much in place, but there are 
fundamental questions on which it is proper to make up 
one's mind." 

These remarks appear to me to be well founded, and I 
must state why it is that I have ventured to publish an 
extensive work on logic, without properly making up my 
mind as to the fundamental- nature of the reasoniuf 
process. The fault after all is one of omission mther than 
of commission. It is open to me on a future occasion to 
supply the deficiency if I should ever feel able to under- 
take the task. P>ut I do not conceive it to be an essential 
part of any treatise to enter into an ultimate analysis of 
its subject matter. Analyses must always end somewhere. 
There were good treatises on light which described the 
laws of the phenomenon correctly before it was known 
whether light consisted of undulations or of projected 
particles. Now we have treatises on the Undulatory 
Tlieory which are very valuable and satisfactory, although 
they leave us in almost complete doubt as to what the 
vibrating medium really is. So I think that, in the 
present day, we need a correct and scientific exhibition 
of the formal laws of thought, and of the forms of 
reasoning based on them, although we may not be able 
to enter into any complete analysis of the nature of those 
laws. What would the science of geometry be like now 
if the Greek geometers had decided that it was improper 
to publish any propositions before they had decided on 

the nature of an axiom? Where would the science of 
aiithmetic be now if an analysis of the nature of number 
itself were a necessary preliminary to a development of 
the results of its laws ? In recent times there have been 
enormous additions to the mathematical sciences, but very 
few attempts at psychological analysis. In the Alex- 
andrian and early mediajval schools of philosophy, much 
attention was given to the nature of unity and plurality 
chiefly called forth by the question of the Trinity. In 
the last two centuries whole sciences have been created 
out of the notion of plurality, and yet speculation on the 
nature of plurality has dwindled away. This present 
treatise contains, in the eighth chapter, one of the few 
recent attempts to analyse the notion of number itself. 

If further illustration is needed, I may refer to the 
differential calculus. Nobody calls in question the formal 
truth of the results of that calculus. All the more exact 
and successful parts of physical science depend upon its 
use, and yet the mathematicians who have created so 
great a body of exact truths have never decided upon 
the basis of the calculus. What is the nature of a limit 
or the nature of an infinitesimal? Start the question 
among a knot of mathematicians, and it will be found 
that hardly two agree, unless it is in regarding the question 
itself as a trifling one. Some hold that there are no such 
things as infinitesimals, and that it is all a question of 
limits. Others would argue that the infinitesimal is the 
necessary outcome of the limit, but various shades of 
intermediate opinion spring up. 

Now it is just the same with logic. If the forms of 
deductive and inductive reasoning given in the earlier 
part of this treatise are correct, they constitute a definite 
addition ta logical science, and it would have been absurd 
to decline to publish such results because I could not at 
the same time decide in my own mind about the psy- 
chology and philosophy of the subject. It comes in short 
to tliis, that my book is a book on Formal Logic and 



.^ (' 



Scientific Method, and not a book on psychology and 

It may be objected, indeed, as the Spectator objects, 
that Mill's System of Logic is particularly strong in the 
discussion of the psychological foundations of reasoning, 
so that Mill would appear to have successfully treated 
that which I feel myself to be incapable of attempting at 
present. If Mill's analysis of knowledge is correct, then 
I have nothing to say in excuse for my own deficiencies. 
But it is well to do one thing at a time, and therefore 
I have not occupied any considerable part of this book 
with controversy and refutation. What 1 have to say of 
Mill's logic will be said in a separate work, in which 
his analysis of knowledge will be somewhat minutely 
analysed. It will then be shown, I believe, that Mill's 
psychological and philosophical treatment of logic has not 
yielded such satisfactory results as some writers seem to 

Various minor but still important criticisms were made 
by Professor Robertson, a few of which have been noticed 
in the text (pp. 27, loi). In other cases his objections 
hardly admit of any other answer than such as consists 
in asking the reader to judge between the work and the 
criticism. Thus Mr. Robertson asserts* that the most 
complex logical problems solved in this book (up to p. 102 
of this edition) might be more easily and shortly dealt 
with upon the principles and with the recognised methods 
of the traditional logic. The burden of proof here Jies 
upon Mr. Robertson, and his only proof consists in a 
single case, where he is able, as it seems to me accidentally, 
to get a special conclusion by the old form of dilemma. 
It would be a long labour to test the old logic upon every 
result obtained by my notation, and I must leave such 

* Portions of this work, have already been pablished in my articles 
entitled "John Stuart Mill's Philosophy Tested," printed in the ConUm- 
porary lUview for December, 1877, vol. xxxi. p. 167, and for January anri 
Apnl, 1878, vol. xxxL p. 256, and vol. xxxii. p. 88. (Nolo added iu 
November, 1878.) « Mind, vol. L p. 222 



readers as are well acquainted with the syllogistic logic to 
pronounce upon the comparative simplicity and power of 
the new and old systems. For other acute objections 
brought forward by Mr. Robertson, I must refer the reader 
to the article in question. 

One point in my last chat)ter, that on the Results and 
Limits of Scientific Method, has been criticised by 
Professor W. K. Clifford in his lecture 1 on " The First 
and the Last Catastrophe." In vol. ii. p. 438 of the 
first edition (p. 744 of this edition) I referred to certain 
inferences drawn by eminent physicists as to a limit to 
the antiquity of the present order of things. " According 
to Sir W. Thomson's deductions from Fourier's theory of 
heat, we can trace down the dissipation of heat by con- 
duction and radiation to an infinitely distant time when 
all things will be uniformly cold. But we cannot similarly 
trace the Heat-liistory of the Universe to an infinite 
distance in the past. For a certain negative value of tlie 
time, the formulso give impossible values, indicating that 
there was some initial distribution of heat which could 
not have resulted, according to known laws of nature, 
from any previous distribution." 

Now according to Professor Clifford I have here mis- 
stated Thomson's results. "It is not according to the 
known laws of nature, it is according to the known laws 
of conduction of heat, that Sir William Thomson is speak- 
ing. ... All these physical writers, knowing what they 
were writing about, simply drew such conclusions from 
the facts which were before them as could be reasonably 
drawn. They say, here is a state of things which could 
not have been produced by the circumstances we are at 
present investigating Then your speculator comes, he 
reads a sentence and says, ' Here is an opportunity for 
me to have my fling.' And he has his fling, and makes a 
purely baseless theory about the necessary origin of the 

nJ. ^'T^^'^d^tly Review New Series. April 1875. p. 480. Lecture ro- 
printed by the Sunday Lecture Society, p. 24. 



J\ .!^' 



present order of nature at some definite point of timo, 
which might be calculated." 

Professor Clifford proceeds to explain that Thomson's 
formulfB only give a limit to the heat history of, say, the 
earth's cnist in the solid stat^. We are led back to the 
time when it became solidified from the fluid condition. 
There is discontinuity in the histoiy of the solid matter, 
but still discontinuity which is within our comprehension. 
Still further back we should come to discontinuity again, 
when the liquid was formed by the condensation of heated 
gaseous matter. Beyond that event, however, there is 
no need to suppose further discontinuity of law, for the 
gaseous matter might consist of molecules which had been 
falling together from different parts of space through infinite 
past time. As Professor Clifford says (p. 481) of the 
bodies of the universe, " What they have actually done 
is to fall together and get solid. If we shoiild reverse 
the process we should see them separating and getting 
cool, and as a limit to that, we should find that all these 
bodies would be resolved into melecules, and all these 
would be flying away from each other. There would be 
no limit to that process, and we could trace it as far back 
as ever we liked to trace it." 

Assuming that I have erred, I should like to point out 
that I have erred in the best company, or more strictly, 
being a speculator, I have been led into error by the best 
physical writers. Professor Tait, in his Sketch of Ther- 
modynamics, speaking of the laws discovered by Fourier 
for the motion of heat in a solid, says, " Their mathematical 
expressions point also to the fact that a uniform distribu- 
tion of heat, or a distribution tending to become uniform, 
must have arisen from some primitive distribution of heat 
of a kind not capable of being produced by known laws 
from any previous distribution." In the latter words it 
will be seen that there is no limitation to the laws of 
conduction, and, although I had carefully refeiTed to 
Sir W. Thomson's original paper, it is not unnatural 



that I should take Professor Tait's interpretion of its 

In his new work On some, Recent Advances in Physical 
Science^ Professor Tait has recurred to the subject as 
follows : * " A profound lesson may be learned from one 
of the earliest little papers of Sir W. Thomson, published 
while he was an undergraduate at Cambridge, where he 
shows that Fourier's magnificent treatment of the con- 
duction of heat [in a solid body] leads to formulae for its 
distribution which are intelligible (and of course capable 
of being fully verified by experiment) for all time future, 
but which, except in particular cases, when extended to 
time past, remain intelligible for a finite period only, and 
then indicate a state of things which could not have 
resulted under known laws from any conceivable previous 
distribution [of heat in the body]. So far as heat is 
concerned, modern investigations have shown that a 
previous distribution of the ma^/<jr involved may, by its 
potential energy, be capable of producing such a state of 
things at the moment of its aggregation ; but the example 
is now adduced not for its bearing on heat alone, but as 
a simple illustrntion of the fact that all portions of our 
Science, especially that beautiful one, the Dissipation 
of Energy, point unanimously to a beginning, to a state of 
things incapable of being derived by present laws [of 
tangible matter and its energy] from any conceivable 
previous arrangement." As this was published nearly a 
year after Professor CUfford's lecture, it may be infeiTed 

Txli!';^ Thomson's words are as follows {CamhHdge Mathematical 
ZrZl:, r- '^^,^'J°^- »»• ^' ,^74). "When x is ne^tivo, the state 
[f.L wM ? T''''^ ^. *^? '^""^^ ^^ Siny possible distribution of tempera- 
Ho^pn^« f \^ ^, P^^^^o'^-'ly existed." There is no limitation in the 
St« of ll f ?-'^' *?^ conduction, but, as the whole paper treats of the 
i^*T.V l?^°l 1- **''" o "" ",^^'^' '^ "'^y »° <^o"^t be understood that there 

ourn^fnr PM "• ^^S ^^° t ?"^°"^ l^^P^'- «« the subject in the same 
KllmiSr'^' '^ '''- "• ^- ''' ^''^'^ ^eain ^here is no ex- 

Tait'8^coSo^1•on.'^n^l^*"'"u'f ^' "'^ ^" ^^'^ original, and show Professor 
ti^tlf'a^^Tn oT^^^ ',68.r^'^" "'^''^ '' '^^ ''''''''''' ''''' «"^j-^ » 





that Professor Tait adheres to his original opinion that 
the theory of heat does give evidence of " a beginning." 

I may add that Professor Clerk Maxwell's words seem 
to countenance the same view, for he says,^ " This is only 
one of the cases in which a consideration of the dissi- 
pation of energy leads to the determination of a superior 
limit to the antiquity of the observed order of things." 
The expression " observed order of things " is open to 
much ambiguity, but in the absence of qualification I 
should take it to include the aggregate of tlie laws of 
nature known to us. I should interpret Professor Maxwell 
as meaning that the tlieory of heat indicates the occuiTence 
of some event of which our science cannot give any 
further explanation. The physical writers thus seem not to 
be so clear about the matter as Professor Clifford assumes. 

So far as I may venture to form an independent 
opinion on the subject, it is to the eflect that Professor 
Clifford is right, and that the known laws of nature do 
not enable us to assign a " beginning." Science leads us 
backwards into infinite past duration. But that Professor 
Clifford is right on this point, is no reason why we should 
suppose him to be right in his other opinions, some of 
which I am sure are wrong. Nor is it a reason why other 
parts of my last chapter should be wiong. The question 
only affects the single paragraph on pp. 744-5 of this 
book, which might, I believe, be struck out without 
necessitating any alteration in the rest of the text. It 
is always to be remembered that the failure of an argu- 
ment in favour of a proposition does not, generally 
speaking, add much, if any, probability to the contra- 
dictory proposition. I cannot conclude without expressing 
my acknowledgments to Professor Clifford for his kind 
expressions regarding my work as a whole. 

* Theory of Heat, 1871, p. 24$. 

2, Th« Chestnuts, 
West Heath, 

Hamistead, N.W. 
August IS, 1877. 






1. Introduction j 

2. The Powers of Miud concerned in the Creation of Science . . 4 

8. Laws of Identity and Difference 5 

4. The Nature of the Laws of Identity and Difference . . . .* 6 

6. The Process of Inference 9 

6. Deduction and Induction ] n 

7. Symbolic Expression of Logical Inference 13 

8. Kxpresaion of Identity and Dillerence 14 

9. General Formula of Logical Inference .17 

10. The Propagating Power of Similarity ! * *. 20 

U. Anticipations of the Principle of Substitution ... ' . .* 21 
12. The Logic of Kelatives . . ! ! 22 



1. Term* 

2. Twofold mear.ini? of (Jenend Namen * ' «k 

8. Abstract Terms .... ' ^ 

4. Subauuti*; Terms . . \ [ '. \ \ . [ ' ' ' ^ 

















5. Collective Tenus 

6. Synthesis of Terms 

7. Symbolic Expression of tbe Lnw of Contradiction 

8. Certain Special Conditions of Logical Symbols . 



. 29 

. 80 
. 81 
. 82 



Propositions . . . 
Simple Identities . 
Partial Identities . . 
Limited Identities . 
Negative rro]K)sitions 
Conversion of Propositions 


7. Twofold Interpretation of Fropcutionw 



1. Deductive Reasoning 49 

2. Immediate Inference 50 

8. Inference with Two Simple Identities 51 

4. Inference with a Simple and a Paiaial Identity 53 

5. Inference of a Partial from Two Partial Identities ... .55 

6. On the Ellipsis of Terms in Partial Identities 57 

7. Inference of a Simple from Two Partial Identities 58 

8. Inference of a limited from Two Partial Identities .... 59 

9. Miscellaneous Forms of Detluctive Inference .... . . CO 

*jO. Fallacies 62 




1. Disjunctive Propositions 66 

2. Expression of the Alternative Relation 67 

8. Nature of the Alternative Relation 68 

4. Laws of the Disjunctive Relation 71 

5. Symbolic Expression of the Law of Duality 73 

6. Various Forms of the Disjunctive Proposition 74 

7. Inference by Disjunctive Propositioua 76 






1. The Indirect Method of Inference 81 

2. Simple Illustrations 83 

3. Employment of the Contrapositive Proposition 84 

4. Contrapositive of a Simple Identity 86 

5. Miscellaneous Examples of the Method 88 

6. Mr. Venn's Problem 90 

7. Abbreviation of the Process 91 

8. The Logical Alphabet 94 

9. The Logical Slate 95 

10. Abstraction of Indifferent Circumstances 97 

11. Illustrations of the Indirect Method 98 

12. Second Example 99 

13. Third Example 100 

14. Fourth Example 101 

15. Fifth Example 101 

16. Fallacies Analysed by the Indirect Method 102 

17. The logical Abacus 104 

18. The L<^cal Machine 107 

19. The Order of Premises 114 

20. The Equivalence of Propositions ! . 115 

21. The Nature of Inference 118 



1. Induction 121 

2. Induction an Inverse Operation 122 

3. Inductive Problems for Solution by the Reader ! 126 

4. Induction of Simple Identities ! .* 127 

5. Induction of Partial Identities ' 130 

6. Solution of the Inverse or Inductive Problem, involving Two 

Classes ^ ^ 134 

7. The Inverse Logical Problem, involving Three Classes . . ] 137 

8. Professor Clifford on the Types of Compound Statpaiar.t in- 

volving Four Classes 143 

9. Distinction between Perfect and Imperfect Induction' .* .' .' 146 
10. Transition from Perfect to Imperfect Induction 149 







■aonON PAGE 

6. Comparison of the Theory with Experience 206 

6. Probable Deductive Arguments 209 

7. Difficulties of the Theory 213 

I / 




[} ■ (^ 





1. Princn>Ie8 of Number 153 

2. The Nature of Number ifig 

8. Of Numerical Abstraction 158 

4. Concrete and Abstract Number 159 

5. Analogy of Logical and Numerical Terms 160 

6. Principle of Mathematical Inference . . 162 

7. Reasoning br Inequalities 165 

8. Arithmetical Reasoning , 167 

9. Numerically Definite Reasoning 168 

10. Numerical meaning of Logical Conditioua 171 




1. The Variety of Nature 178 

2. Distinction of Combinations and Permutations .177 

8. Calculation of Number of Combinations 180 

4. The Arithmetical Triangle 182 

5. Connexion between the Arithmetical Triangle and the logical 

Alphabet 189 

C. Pctiidble Variety of Nature and Ark 190 

7. Higher Orders of Variety 192 



1. Philosophy of Inductive Inference ." . . . 218 

2. Various Classes of Inductive Truths ... 219 

8. The Relation of Cause and Effect 220 

4. Fallacious Use of the Term Cause 221 

5 Confusion of Two Questions , . 222 

6. Definition of the Term Cause 224 

7. Distinction of Inductive and Deductive Results 220 

8. The Grounds of Inductive Inference 228 

9. Illustrations of the Inductive Process 229 

10. Geometrical Reasoning 283 

11. Discrimination of Certainty and Probability ..!.!*. 235 




1. The Inductive or Inverse Application of the Theory . . 240 

2. Pnnciple of the Inverse Method . . * 242 

3. Simple Applications of the Inverse Method '. 244 

i' m?® Theory of Probability in Astronomy. ...*.'*" 247 

5. The General Inverse Problem " * 050 

6. Simple Illustration of the Inverse Problem . . . [ ' 258 

7. General Solution of the Inverse Problem. ...!*.** 255 

8. Rules of the Inverse Method * * ' 257 

9. Fortuitous Coincidences ......* 261 

10. Summary of the Theory of Inductive Inference .*;.** *. 266 



1. Theory of Probability 

2. Fundunental Principles of the Theory . . . 
8. Rules for the Galenlatioii of ProbabUities . . 
4. i'he Lutpcai Al|»habet in queetioas of Prvbability 

. 197 
. 200 
. 208 
. S06 










BOOK ni. 




1. The Exact Measurement of Phenomena . . . . ^ . . 270 

2. Division of the Subject • 274 

8. Continuous quantity ..../• 274 

4. The Fallacious Indications of the Sensen .... . 276 

5. Complexity of Quantitative Questions . 278 

6. The Methods of Accurate Mftianromont .... . 282 

7. Conditions of Accurate Measurement 282 

8. Measuring Instruments .... 284 

9. The Method of Repetition 288 

10. Measurements by Natural Coincidence 292 

11. Modes of Indirect Measuiei/ieAt 296 

12. Comparative Use of Measuring Instruments 299 

13. Systematic Performance of Meaaurementa 800 

14. The Pendulum 302 

15. Attainable Accuracy of Measurement 30S 




1. Analyaia of Quantitative Phenomena 335 

2. lUuitrations of the Com^ication of Etfects 336 

8. Methods of Eliminating Error 839 

4. Method of Avoidance of Error .......... 340 

6. Differential Method 844 

6. Method of Correction 846 

7. Method of Compensation 350 

8. Method of Reversal 854 




The Method of Means 357 

Several Uses of the Mean Result 359 

The Mean and the Average 3(j0 

On the Average or Fictitious Mean ... ..... 368 

The Precise Mean Result ' , . . 866 

Determination of the Zero Point 363 

Determination of Maximum Points . . . . . 8/1 












Units and Standards of Measurement ... 305 

Stan.iard Unit of Time 807 

The Unit of Space and the Bar Standard 312 

The Terrestriiil Standard 814 

The Pendulum Standard 315 

Unit of Density 316 

Unit of Mass .317 

Natural System of Standards 319 

Subsidiary Units 320 

Derived Units 321 

Provisional Units 823 

Theory of Dimensions .... 825 

Natural Constants ... 328 

Mathematical Constants . , . . 880 

Physical Constants ... ... 881 

Astrunomioal Constants . ... 88S 

Terrestrial Numbers . ..... 888 

Oiganic Numbers ... . •ijjS 

Social Numbers ... • • • "• 



1. The Law of Error 374 

2. Establishment of the Law of Error . . . . . . . . 375 

8. Herschel's Geometrical Proof ....*...** ' 377 

4. lAplace's and Quetelet's Proof of the Law . . .* * * 378 

o. Logical Origin of the Law of Error .... 383 

6. Verification of the Law of Error . . 333 

7. The Probable Mean Result .,...!...' i * 385 

8. The Probable Error of Results ...*.*....!* 386 

9. R«jeotion of the Mean Result . . ! ago 

10. Method of Least Squares . . J ' 

lo y*!'^.^P***i*^^'^^''«'"y of Probability .;;*..'. .* 894 
VL Detection of Constant Errors ....... 3M 





1^ ii 






1. Observation . 899 

2. Distinction of Observation and Experiment . . . • 400 

3. Mental Conditions of Correct Observation 402 

4. InstrumeDtal and Sensual Conditions of Correct Observation . 404 
6. External Conditions of Correct Observation 407 

6. Apparent Sequence of Events 409 

7. Negative Arguments from Non^Obaervation 411 



1. Experiment 410 

2. Exclusion of Indifferent CircumstanceA 419 

3. Simplification of Experiments 422 

4. Failure in the Simplification of Experiments 424 

5. Removal of Usual Conditions 426 

6. Interference of Unsuspected Conditions 428 

7. Blind or Test Experiments 488 

8. Negative Results of Experiment .... ... . 434 

9. Limits of Experiment . 437 






1. Theory of Approximation ... 466 

2. Substitution of Siinple hypotneses 458 

3. Approximation to Exact Laws 462 

4. Successive Approximations to Natural Conditions 465 

5. Discovery of Hypothetically Simple Laws 470 

6. Mathematical Principles of Approximation 471 

7. Approximate Independence of Small Effects 476 

8. Four Meanings of Equality 479 

9. Arithmetic of Approximate Quantities 481 



1. Quantitative Induction . ■ 483 

2. F^bable Connexion of Varying Quantities .... . 484 

3. Empirical Mathematical Laws 487 

4. Discovery of Rational Formulae 489 

6. The Graphical Method 492 

6. Interpolation and Extrapolation 495 

7. Illustrations of Empirical Quantitative I.Hwa 499 

8. Simple Proportional Variation .... 601 




MBiHoo or rA&lAllOXIk 

1. Method of Variations . . 489 

2. The Variable and the Variant 440 

3. ^leasurement of the Variable 441 

1-. Maintenance of Similar Conditions 443 

6. Collective Experiments 445 

6. Periodic Variations 447 

7. Combined Periodic Changes 450 

8. Principle of Forced Vibrations 451 

9. Integrated Variations . . . . , 468 

1. The Use of Hypothesis , 504 

2. Requisites of a good Hyjwthesis , 510 

3. Possibility of Deductive Reasoning 511 

4. Consistency with the Laws of Nature 614 

6. Conformity with Facts ,,...! 516 

6, Experimentum Crucis 1 *, ! 518 

7. Descriptive Hypotheses ! ! * * 522 









1. Empirical Knowledge, Explanation and Prediction .... 525 

2. Empirical Knowledge 526 

8. Accidental Discovery 529 

4. Empirical Observations subsequently Explained 532 

5. Overlooked Results of Theory 534 

6. Predicted Discoveries 536 

7. Predictions in tlie Science of Light 588 

8. Predictions from the Theory of Undulations 540 

9. Prediction in other Sciences 542 

10. Prediction by Inversion of Cause and Effect 545 

11. Facts known only by Theory 547 



1. Accordance of Quantitative Theories ... 551 

2. Empirical Measurements 552 

8. Quantities indicated by Theory, bat Empirically Measured . . 553 

4. Explained Results of Measurwnent 554 

5. Quantities determined by Theory and verified by Measurement 555 

6. Quantities determined by Theory and not verified 556 

7. Discordance of Theory and Experiment 558 

8. Accordance of Measurements of Astronomical Distances . . 560 

9. Selection of the best Mode of Measurement 563 

10. Agreement of Distinct Modes of Measurement 564 

11. Residual Phenomena 569 



1. Character of the Experimentalist ...» 574 

2. Error of the Baconian Method 576 

3. Freedom of Theorising 577 

4. The Newtonian Methcni, the True Organum 681 

5. Candour and Courage of the Philosophic Mind 586 

6. The Philosophic Character of Faraday 687 

7. Reservation of Judgment .... ...... 592 





1. Generalisation .... 594 

2. Distinction of Generalisation and Analog\' . 596 

3. Two Meanings of Generalisation 597 

4. Value of Generalisation . . . 599 

5. Comparative Generality of Properties 600 

6. Uniform Properties of all Matter 603 

7. Variable Properties of Matter 606 

8. Extreme Instances of Properties 607 

9. The Detection of Continuity 610 

10. The Law of Continuity 615 

11. Failure of the Law of Continuity 619 

12. Negative Alignments on the Principle of Continuity .... 621 

13. Tendency to Hasty Grcneralisation 623 



1. Analogy g27 

2. Analogy as a Guide in Discoveiy 629 

3. Analogy in the Mathematical Sciences . . . . . . . .' 631 

4. Analogy in the Theory of Undulations 635 

5. Analogy in Astronomy 533 

6. Failures of Analogy 641 



1. Exceptional Phenomena q^^ 

2. Imaginary or False Exceptions .....!...* 647 

3. Apparent but Congruent Exceptions . . ^ ! . G49 

4. Singular Exceptions gro 

5. Divergent Exceptions ] g^g 

6. Accidental Exceptions '. . . . . ' 658 

7. Novel and Unexplained Exceptions' ! . 661 

8. Limiting Exceptions 553 

9. Real Exceptions to Supposed Laws* . [ . aab 

10. Unclassed Exceptions .W^. . i i .' ! 668 




» P| 



■f >'' li 





1. Classificatioii 673 

2. Classification involving Induction . 675 

3. Multiplicity of Modes of Classification . 677 

4. Natural and Artificial Systems of Classification 679 

5. Correlation of Properties 681 

6. Classification in Crystallography . 685 

7. Classification an Inverse and Tentative Operation 689 

8. Symbolic Statement of the Theory of Classification .... 692 

9. Bifurcate Classification 694 

10. The Five Predicablcs 698 

11. Summum Genus and Infima Species ... 701 

12. The Tree of Porphyry 702 

13. Does Abstractiou imply Generalisation ? 704 

14. Discovery of Marks or CharacterisU'cs 708 

15. Diagnostic Systems of Classification ... 710 

16. Index Classifications 714 

17. Classification in the Biological ScienoM 718 

18. Classification by Types 722 

19. Natural Genera and Species ... 724 

20. Uui(^ue or Exceptional Objects 728 

21. Limits of Classification 730 




Reflections on the Results and Limits of Scientific Method . . 735 

The Meaning of Natural Law 787 

Infiniteness of the Universe 788 

The Indeterminate Problem of Creation 740 

Hierarchy of Natural Laws 742 

The Ambiguous Expression — *• Uniformity of Nature "... 745 

Possible States of the Universe 749 

S|)eculation8 on the Keconcentration of Energy 751 

The Divergent Scope for New Discovery 752 

Infinite Incompleteness of the Mathematical Sciences . . . 754 
The Reign of Law in Mental and Social Phenomena . . . .759 

The Theory of Evolution 761 

Possibility of Divine Interference .... 765 









14. Conclnsion 







Science arises from the discovery of Identity amidst r\ 
Diversity. The process may be described in different ^ 
words, but our language must always imply the presence 
of one common and necessary element. In every act of 
inference or scientific method we ai-e engaged about a 
certain identity, sameness, similarity, likeness, resemblance, 
analogy, equivalence or equality apparent between two 
objects. It is doubtful whether an entirely isolated 
phenomenon could present itself to our notice, since there 
must always be some points of similarity between object 
and object. But in any case an isolated phenomenon 
could be studied to no useful purpose. The whole value 
of science consists in the power which it confers upon 
us of applying to one object the knowledge acquired 
from like objects ; and it is only so far, therefore, as we can 
discover and register resemblances that we can turn our 
observations to account. 

. -^^^^^ ^ * spectacle continually exhibited to our senses, 
in which phenomena are mingled in combinations of 
endless variety and novelty. Wonder fixes the mind's 
attention ; memory stores up a record of each distinct 
impression ; the powers of association bring forth the record 
when the like is felt again. By the higher faculties of 
juagment and reasoning the mind compares the new with 





! i 




the old, recognises essential identity, even when disguised 
by diverse circumstances, and expects to find again what 
was before experienced. It must be the ground of all 
reasoning and inference that what is true of one thing unit 
be IrvLe of its equivalent, and that under carefully ascertained 
conditions Nature repeats herself 

Were this indeed a Chaotic Universe, the powers of mind 
employed in science would be useless to us. Did Chance 
wholly take the place of order, and did all phenomena 
come out of an Infinite Lottery, to use Condorcet's ex- 
pression, there could be no reason to expect the like result 
in like circumstances. It is possible to conceive a world 
in which no two things should be associated more often, in 
the long run, than any other two things. The frequent 
conjunction of any two events would then be purely 
fortuitous, and if we expected conjunctions to recur con- 
tinually, we should be. disappointed. In such a world we 
might recognise the same kind of phenomenon as it ap- 
peared from time to time, just as we might recognise a 
marked ball as it was occasionally drawn and re-drawn 
from a ballot-box ; but the approach of any phenomenon 
would be in no way indicated by what had gone before, 
nor would it be a sign of what was to come after. In such 
a world knowledge would be no more than the memory of 
past coincidences, and the reasoning powers, if they existed 
at all, would give no clue to the nature of the present, and 
no presage of the future. 

Happily the Univei-se in which we dwell is not the 
result of chance, and where chance seems to work it is 
our own deficient faculties which prevent us from recog- 
nising the operation of Law and of Design. In the material 
framework of this world, substances and forces jiresent 
themselves in definite and stable combinations. Things 
are not in perpetual flux, as ancient philosophers hem. 
Element remains element; iron changes not into gold. 
With suitable precautions we can calculate upon finding 
the same thing again endowed with the same properties. 
The constituents of the globe, indeed, appear in almost 
endless combinations ; but each combination bears its fixed 
character, and when resolved is found to be the compound 
of definite substances. Misapprehensions must continually 
occur, owing to the limited extent of our exi)erieuce. We 



can never have examined and registered possible exist- 
ences so thoroughly as to be sure that no new ones will 
occur and frustrate our calculations. The same outward 
appearances may cover any amount of hidden differences 
which we have not yet suspected. To the variety of sub- 
stances und powers diffused through nature at its creation, 
we should not suppose that our brief experience can assign 
a limit, and the necessary imperfection of our knowledge 
must be ever borne in mind. 

Yet there is much to give us confidence in Science. Tho 
wider our experience, the more minute our examination of 
the globe, the greater the accumulation of well-reasoned 
knowledge, — the fewer in all probability will be the failures 
of inference compared with the successes. Exceptions 
to the prevalence of Law are gradually reduced to Law 
themselves. Certain deep similarities have been detected 
among the objects around us, and have never yet been 
found wanting. As the means of examining distant parts 
of the universe have been acquired, .those similarities have 
been traced there as here. Other worlds and stellar 
systems may be almost incomprehensively different from 
ours in magnitude, condition and disposition of parts, and 
yet we detect there the same elements of which our own 
limbs are composed. The same natural laws can be 
detected in operation in every part of the universe within 
the scope of our instruments ; and doubtless these laws are 
obeyed irrespective of distance, time, and circumstance. 

It is the prerogative of Intellect to discover what is uni- 
form and unchanging in the phenomena around us. So 
far as object is different from object, knowledge is useless 
and inference impossible. But so far as object resembles 
object, we can pass from one to the other. In proportion 
as resemblance is deeper and more general, the com- 
manding powers of knowledge become more wonderful. 
Identity in one or other of its phases is thus always 
the bridge by which we pass in inference from case to 
case ; and it is my purpose in this treatise to trace out the 
various forms in which the one same process of reasoning 
presents itself in the ever-growing achievements of Scientific 

B 2 







The Powers of Mind concerrud in the OrecUion of Science, 

It is no part of the purpose of this work to investigate the 
nature of mind. People not uncommonly suppt»se that 
logic IS a branch of psychology, because reasoning is a 
mental operation. On the same ground, however, we 
might argue that all the sciences are branches of psy- 
chology. As will be further explained, I adopt the opinion 
of Mr. Herbert Spencer, that logic is really an objective 
science, like mathematics or mechanics. Only in an in- 
cidental manner, then, need I point out that the mental 
powers employed in the acquisition of knowledge are prob- 
ably three in number. They are substantially as Professor 
Bain has stated them i : — 

1. The Power of Discrimination. 

2. The Power of Detecting Identity. 

3. The Power of Retention. 

We exert the first power in every act of perception. 
Hardly can we have a sensation or feeling unless we dis- 
criminate it from something else which preceded. Con- 
sciousness would almost seem to consist in the break 
between one state of mind and the next, just as an induced 
current of electricity arises fiom the beginning or the 
ending of the primary current. We are always engaged in 
cliscnmmation ; and the rudiment of thought which exists 
m the lower animals probably consists in their power of 
feeling difference and being agitated by it. 

Yet had we the power of discrimination only. Science 
could not be created. To know that one feeling differs 
from another gives purely negative information. It cannot 
teacli us what will happen. In such a state of intellect 
each sensation would stand out distinct from every other • 
there would be no tie, no bridge of affinity between them! 
We want a unifying power by which the present and the 
future may be linked to the past ; and this seems to be 
accomplished by a different power of mind. Lord Bacon 
has pointed out that different men possess in very different 
degrees the powers of discrimination and identification. It 
may be said indeed tliat discrimination necessarily implies 
the action of the opposite process of identification ; and so 
Tht Se7ws and the Intellect, Second Ed., pp. 5, 325, &c. 

it doubtless does in negative points. But there is a rare 
property of mmd which consists in penetrating the dis- 
guise of variety and seizing the common elements of 
sameness; and it is this property which furnishes the true 
measure of intellect. The name of" intellect " expresses the 
interlacing of the general and the single, which is the 
peculiar province of mind.i To cogitate is the I^tin co- 
agitate, restmg on a like metaphor. Logic, also is but 
another name for the same process, the peculiar work of 
reason ; for X0709 is derived from X^^^v, which like the 
L&tmlegere meant originally to gather. Plato said of this 
unifying power, that if he met the man who could detect 
t/u one %n the many, he would follow him as a god. 

Laws of Identity and Difference, 

At the base of all thought and science must lie the 
laws which express the very nature and conditions of the 
discriminating and identifying powers of mind These 
are the so-called Fundamental Laws of TJiought, usually 
stated as follows : — & > j 

1. The Law of Identity. Whatever is, is. v/ 

2. The Law of Contradiction. A thing cannot both be 

and not he. 

3. The Law of Duality. A thing must eitUv he m^ 

not he. 

The first of these statements may perhaps be regarded as 
LnTZ>'''f 'i '^'"^^'^ ^^^^^' '^ '' fundamental a notion 

fectlt h' r ^''fiP-V^t "^."""^ ^^ ^°^ «^^^ent is peiv 
fectly Identical with itself, and, if any pei^on were unaware 

desorihT^r^ f ^^' r'^ " '^'""^^y'' ^^ ^^^1^ ^«t better 
ae^ribe it than by such an example. 

rJnofT'^-^^'^?''^''^?"^ ^^^^ contradictory attributes 
T^^.l'J'^ ^T^^ ^Sether. The same object may vary 

wVte . ^f r^^V^^'^'J ^'^'\'^ '''^y ^^ ^l^^k' and there 
White, at one time it may be hard and at another time 

ToLar6^^"o?'sfxtt^^ ^ Scimce 0/ Language, Second Series, 
A'i(^hne;n?n'cr of « ^f^ll /- °' .^^^- "; P' ^7- The view of the etymolo 
mTx m^y I intellect" is g.ren above on the authority of Professor 

to wh^ch h« lit''-";'/;- ^' 'PP^'"^ "^ *^^ ''"^'''^'y ^Pi»^on, accoSg 
<^<I l^wiln Y'a- ''¥^}^''' ?"^^n« to choose between, to s^e a ditfei? 
«ice between, to discrnnmate, instead of to unite. 



M ■!i 



soft ; but at the same time and place an attribute cannot be 
both present and absent Aristotle truly described this 
law as the first of all axioms— one of which we need not 
seek for any demonstration. All truths cannot be proved, 
otherwise there would be an endless chain of demonstration ; 
and it is in self-evident truths like this that we find the 
simplest foundations. 

The third of these laws completes the other two. It 
asserts that at every step there are two possible alter- 
natives—presence or absence, affirmation or negation. 
Hence I propose to name this law the Law of Dualfty, for 
it gives to all the formulae of reasoning a dual character. It 
asserts also that between presence and absence, existence 
and non-existence, affirmation and negation, there is no 
third alternative. As Aristotle said, there can be no mean 
between opposite assertions: we must either affirm or 
deny. Hence the inconvenient name by which it has been 
known— The Law of Excluded Middle. 

It may be allowed that these laws are not three indepen- 
dent and distinct laws ; they rather express three different 
aspects of the same truth, and each law doubtless pre- 
supposes and implies the other two. But it has not 
hitherto been found possible to state these characters of 
identity and difference in less than the threefold formula. 
The reader may perhaps desire some information as to the 
mode in which these laws have been stated, or the 
way in which they have been regarded, by philosophers 
in different ages of the world. Abundant information 
on this and many other points of logical history will be 
found in Veherweg's Sf/stem of Logic, of which an excellent 
translation has been published by Professor T. M. Lindsay 
(see pp. 228-281). 

The Nature of the Laws of I<kntity and Difference, 

I must at least allude to the profoundly difficult ques- 
tion concerning the nature and authority of these Laws of 
Identity and Difference. Are they Laws of Thought or 
Laws of Things ? Do they belong to mind or to material 
nature ? On the one hand it may be said that science is a 
purely mental existence, and must therefore conform to the 
laws of that which formed it. Science is in the mind and 


not in the things, and the properties of mind are therefore 
all important It is true that these laws are verified in the 
observation of the exterior world ; and it would seem that 
they might have been gathered and proved by general- 
isation, had they not already been in our possession. But 
on the other hand, it may well be urged that we cannot 
prove these laws by any process of reasoning or observation, 
because the laws themselves are presupposed, as Leibnitz 
acutely remarked, in the very notion of a proof. They are 
the prior conditions of all thought and all knowledge, and 
even to question their truth is to allow them true. Hartley 
ingeniously refined upon this argument, remarking that if 
the fundamental laws of logic be not certain, there must 
exist a logic of a second order whereby we may determine 
the degree of uncertainty : if the second logic be not certain, 
there must be a third ; and so on ad infinitum. Thus we 
must suppose either that absolutely certain laws of thought 
exist, or that there is no such thing as certainty whatever.^ 

Logicians, indeed, appear to me to have paid insufficient 
attention to the fact that mistakes in reasoning are always 
possible, and of not unfrequent occurrence. The Laws 
of Thought are often called necessary laws, that is, laws 
which cannot but be obeyed. Yet as a matter of fact, who 
is there that does not often fail to obey them ? They are 
the laws which the mind ought to obey rather than what 
it always does obey. Our thoughts cannot be the criterion 
of truth, for we often have to acknowledge mistakes in 
arguments of moderate complexity, and we sometimes only 
discover our mistakes by collision between our expectations 
and the events of objective nature. 

Mr. Herbert Spencer holds that the laws of logic are 
objective laws,* and he regards the mind as being in 
a state of constant education, each act of false reasoning 
or miscalculation leading to results which are likely to 
prevent siraUar mistakes from being again committed. 
I am quite inclined to accept such ingenious views ; but 
at the same time it is necessary to distinguish between the 
accumulation of knowledge, and the constitution of the 
'^^d which allows of the acquisition of knowledge. 
Before the mind can perceive or reason at all it must have 

* Hartley on Man, vol. i. p. 359, 

' Frinciples of PsycJwlogy, Second Ed., vol. ii. p. 86. 







the conditions of thought impressed upon it. Before a 
mistake can be committed, the mind must clearly dis- 
tinguish the mistaken conclusion from all other assertions. 
Are not the Laws of Identity and Difference the prior 
conditions of all consciousness and all existence ? Must 
they not hold true, alike of things material and immaterial? 
and if so, can we say that they are only subjectively true 
or objectively true? I am inclined, in short, to regard 
them as true both " in the nature of thought and things/' 
as I expressed it in my first logical essay ; ^ and I hold 
that they belong to the common basis of all existence. 
But this is one of the most difficult questions of psychology 
and metaphysics which can be raised, and it is hardly one 
for the logician to decide. As the mathematician does not 
inquire into the nature of unity and plurality, but develops 
the formal laws of plurality, so the logician, as I conceive, 
must assume the truth of the Laws of Identity and 
Difference, and occupy himself in developing the variety 
of forms of reasoning in which their truth may be 

Again, I need hardly dwell upon the question whether 
logic treats of language, notions, or things. As reasonably 
might we debate whether a mathematician treats of 
symbols, quantities, or things. A mathematician certainly 
does treat of symbols, but only as the instruments 
whereby to facilitate his reasoning concerning quantities ; 
and as the axioms and riiles of mathematical science must 
be verified in concrete objects in order that the calcula- 
tions founded upon them may have any validity or utility, 
it follows that the ultimate objects of matliematical science 
are the things themselves. In like manner I conceive that 
the logician treats of language so far as it is essential for the 
embodiment and exhibition of thought. Even if reasoning 
can take place in the inner cx)nsciousness of man without 
the use of any signs, which is doubtful, at any rate it 
cannot become the subject of discussion until by some 
system of material signs it is manifested to other persons. 
The logician then uses words and symbols as instruments 
of reasoning, and leaves tlie nature and peculiarities of 
language to the grammarian. But signs again must 

* Pure Logic, or the Logic of Quality apart from Quantity, 1864, 
pp- 10, 16, 22, 29, 36, &C. 

correspond to the thoughts and things expressed, in order 
that they shall serve their intended purpose. We may 
therefore say that logic treats ultimately of thoughts and 
things, and immediately of the signs which stand for them. 
Signs, thoughts, and exterior objects may be regarded as 
parallel and analogous series of plienomena, and to treat 
any one of the three series is equivalent to treating either 
of the other series. 

77ie Process of Inference. 

Xiie fundamental action of our reasoning faculties 
consists in inferring or carrying to a new instance of a 
phenomenon whatever we have previously known of its 
like, analogue, equivalent or equal. Sameness or identity 
presents itself in all degrees, and is known under various 
names; but the great rule of inference embraces all 
degrees, and affirms that so far as there exists sameness^ 
identity or likeness, what is true of one thing will he true^ 
of the other. The great difficulty doubtless consists inT 
ascertaining that there does exist a sufficient degree off 
likeness or sameness to warrant an intended inference;! 
and it will be our main task to investigate the conditions 
under which reasoning is valid. In this place I wish to 
point out that there is something common to all acts 
of inference, however different their apparent forms. The 
one same rule lends itself to the most diverse applications. 

The simplest possible case of inference, perhaps, occui-s 
in the use of a pattern, example, or, as it is commonly 
called, a sample. To prove the exact similarity of two 
portions of commodity, we need not bring one portion 
beside the other. It is sufficient that we take a sample 
which exactly represents the texture, appearance, and 
general nature of one portion, and according as this 
sample agrees or not with the other, so will the two 
portions of commodity agree or differ. Whatever is true 
as regards the colour, texture, density, material of the 
sample will be true of the goods themselves. In such 
cases likeness of quality is the condition of inference. 

Exactly the same mode of reasoning holds true of 
magnitude and figure. To compare the sizes of two 
objects, we need not lay them beside each other. A 







staff, string, or other kind of measure may be employed 
to represent the length of one object, and according as it 
agrees or not with the other, so must the two objects 
agree or differ. In this case the proxy or sample represents 
length ; but the fact that lengths can be added and 
multiplied renders it unnecessary that the proxy should 
always be as large as the object. Any standard of 
convenient size, such as a common foot-rule, may be made 
the medium of comparison. The height of a church in 
one town may be carried to that in another, and objects 
existing immovably at opposite sides of the earth may be 
vicanously measured against each other. We obviously 
employ the axiom that whatever is true of a thing as 
regards its length, is true of its equal 

To every other simple phenomenon in nature the same 
principle of substitution is applicable. We may compare 
weights, densities, degrees of hardness, and degrees of all 
other qualities, in like manner. To ascertain whether two 
sounds are in unison we need not compare them directly, 
but a third sound may be the go-between. If a tunin<^- 
fork IS in unison with the middle C of York Minster 
organ, and we afterwards find it to be in unison with the 
same note of the organ in Westminster Abbey, then it 
follows that tl;e two organs are tuned in unison. The 
rule of inference now is, that what is tnie of the tunincr- 
fork as regards the tone or pitch of its sound, is true of 
any sound in unison with it. 

The skilful employment of this substitutive process 
enables us to make measurements beyond the powers of 
our senses. No one can count the vibrations, for instance, 
^\i^A ^**^'^""PiP^- ^"^ we can construct an instrument 
called the siren, so that, while producing a sound of any 
pitch, it shall register the number of vibrations consti- 
tuting the sound. Adjusting the sound of the siren in 
unison with an organ-pipe, we measure indirectly the 
number of vibrations belonging to a sound of that pitch. 
To measure a sound of the same pitch is as good as to 
measure the sound itself. 

Sir David Brewster, in a somewhat similar manner, 
succeeded in measuring the refractive indices of irregular 
fragments of transparent minerals. It was a troublesome, 
and sometimes impracticable work to grind the minerals 




into prisms, so that the power of refracting light could 
be directly observed ; but he fell upon the ingenious device 
of compounding a liquid possessing the same refractive 
power as the transparent fragment under examination. 
The moment when this equality was attained could be 
known by the fragments ceasing to reflect or refract light 
when immersed in the liquid, so that they became almost 
invisible in it. The refractive power of the liquid being 
then measured gave that of the solid. A more beautiful 
instance of representative measurement, depending im- 
mediately upon the principle of inference, could iiot be 

Throughout the various logical processes which we are 
about to consider— Deduction, Induction, Generalisation, 
Analogy, Classification, Quantitative Reasoning— we shall 
find the one same principle operating in a more or less 
disguised foruL 

Deduction and Indicction, 

The processes of inference always depend on the one 
same principle of substitution ; but they may nevertheless 
be distinguished according as the results are inductive or 
deductive. As generally stated, deduction consists in 
passing from more general to less general truths ; induc- 
tion is the contrary process from less to more genera, 
truths. We may however describe the difference in 
another manner. In deduction we are engaged in develop- V 
nig the consequences of a law. We learn the meaning; 
contents, results or inferences, which attach to any given 
proposition. Induction is the exactly inverse process. 
Uven certain results or consequences, we are required to 
discover the general law from which they flow. 

In a certain sense all knowledge is inductive. We can 
only learn the laws and relations of things in nature by 
observing those things. But the knowledge gained from 
the senses is knowledge only of particular facts, and we 
require some process of reasoning by which we may 
collBct out of the facts the laws obeyed by them. 

' Brewster, Treaiiie on New , Philosophical InstrumenU, p. 27^ 
J^ncerning this method see also WheweU, Philosophy of the Inductive 
!^icn, vol. iL p. 355 ; Toinlinson, Philosophical Magazine, Fourth 
^>«ne8, vol xl. p. 328 ; Tyndall, in Younians' Modem Culture, p. 16. 






Experience gives us the materials of knowledge : induction 
digests those materials,, and yields us general knowledge. 
When we possess such knowledge, in the form of 
general propositions and natural laws, we can usefully 
apply the reverse process of deduction to ascertain the 
exact information required at any moment. In its ultimate 
loundation, then, all knowledge is inductive— in the sense 
that it is derived by a certain inductive reasoning from 
the facts of experience. 

It is nevertheless true, — and this is a point to which 
insufficient attention has been paid, that all reasoning 
IS founded on the principles of deduction. I call in 
question the existence of any method of reasoning which 
can be carried on without a knowledge of deductive pro- 
cesses. I shall endeavour to show that induction is really 
the inverse process of deduction. There is no mode of 
ascertaining the laws which are obeyed in certain pheno- 
mena, unless we have the power of determining what 
results would follow from a given law. Just as the 
process of division necessitates a prior knowledge of multi- 
plication, or the integral calculus rests upon the obser- 
vation and remembrance of the results of the differential 
calculus, so induction requires a piior knowledge of 
deduction. An inverse process is the undoing of the 
direct process. A person who enters a maze must either 
trust to chance to lead him out again, or he must carefully 
notice the road by which he entered. The facts furnished 
to us by experience are a maze of particular results; we 
might by chance observe in them the fulfilment of a law, 
but this is scarcely possible, unless we thoroughly learn 
the effects which would attach to any particular law. 

Accordingly, the importance of deductive reasoning is 
doubly supreme. Even when we gain the results of in- 
duction they would be of no use unless we could deduc- 
tively apply them. But before we can gain them at all 
we must understand deduction, since it is the inversion of 
deduction which constitutes induction. Our first task in 
this work, then, must be to trace out fully the nature of 
identity in all its forms of occurrence. Having given any 
series of propositions we must be prepared to develop 
deductively the whole meaning embodied in them, and 
the whole of the consequences which flow from them. 





Symbolic Expression of Logical Inference. 

In developing the results of the Principle of Inference 
we require to use an appropriate language of signs. It 
would indeed be quite possible to explain the processes of 
reasoning by the use of words found in the dictionary. 
Special examples of reasoning, too, may seem to be more 
readily apprehended than general symbolic forms. But it 
has been shown in the mathematical sciences that the 
attainment of truth depends greatly upon the invention of 
a clear, brief, and appropriate system of symbols. Not 
only is such a language convenient, but it is almost 
essential to the expression of those general truths which 
are the very soul of science. To apprehend the truth of 
special cases of inference does not constitute logic ; we 
must apprehend them as cases of more general ''truths. 
The object of all science is the separation of w^hat is 
common and general from what is accidental and different. 
In a system of logic, if anywhere, we should esteem this 
generality, and strive to exhibit clearly what is similar in 
very diverse cases. Hence the great value of general 
symbols by which we can represent the form of a reasoning 
process, disentangled from any consideration of the special 
subject to which it is applied. 

The signs required in logic are of a very simple kind 
As sameness or difference must exist between two things 
or notions, we need signs to indicate the things or 
notions compared, and other signs to denote the relations 
between them. We need, then, (i) symbols for terms, (2) 
a symbol for sameness, (3) a symbol for difference, and (4) 
one or two symbols to take the place of conjunctions. 

Urdmary nouns substantive, such as Iron, Metal, Elec- ^ 
incuy, Undulation, might serve as terms, but, for the 
^ons explained above, it is better to adopt blank letters, 
devoid of special signification, such as A, B. C, Ac! 
Mch letter must be understood to represent a noun, and. 
80 lar as the conditions of the argument allow, any noun, 

iZlfl^' '"^ ^^f"^'. \y> ^> V. ?, &c. are used for any 
El ' ?.^?^t«^\ed or unknown, except when the 
specml conditions of the problem are taken into account, 
80^1 our letters stand for undetermined or unknown 



[oh AT 




:il 1 




These letter-terms will be used indifferently for nouns 
substantive and adjective. Between these two kinds of 
nouns there may perhaps be differences in a metaphysical 
or grammatical point of view. But grammatical usa^re 
sanctions the conversion of adjectives into substantives and 
vice versd; we may avail ourselves of this latitude without 
in any way prejudging the metaphysical difficulties which 
may be involved. Here, as throughout tliis work I sliall 
devote my attention to truths which I can exhibit in a 
clear and formal manner, believing that in the present 
condition of logical science, this course will lead to areater 
advantage than discussion upon the metaphysical questions 
which may underlie any part of the subject. 
^ Every noun or term denotes an object, and usually 
implies the possession by that object of certain qualities 
or circumstances common to all the objects denoted. There 
are certain terms, however, which imply the absence of 
qualities or circumstances attaching to other objects. It 
will be convement to employ a special mode of indicating 
these negative terms, as they are caUed. If the general 
name A denotes an object or class of objects possessing 
certain defined quaUties, then the term Not A will denote 
any object which does not possess the whole of those 
qualities ; m short. Not A is the sign for anything which 
dittere from A m regard to any one or more of the assigned 
qualities. If A denote " transparent object," Not A will 
denote "not transparent object." Brevity and facility of 
expression are of no slight importance in a system of 
notation, and it will therefore be desirable to substitute 
for the negative term Not A a briefer symbol. De Morean 
represented negative terms by small Roman letters or 
sometimes by small italic letters ;i as the latter seeiii to 
be highly convenient, I shall use a, J, c, . . . p, y. &c., aa 
the negative terms corresponding to A, B, C, . . . P, Q, &c. 
Thus if A means " fluid," a wiU mean " not fluid." 

Expression of IderUity and Difference, 

To denote the relation of sameness or identity I unhesi- 
tatingly adopt the sign =, so long used by mathematicians 
to denote equality. This symbol was originally appropriated 

* Formal Logic, p. 38. 


by Robert Recorde in his Whetstone 0/ Wit, to avoid the 
tedious repetition of the words "is equal to;" and he 
chose a pair of parallel lines, because no two things can bo 
more equal.^ The meaning of the sign has however been 
graduaUy extended beyond that of equaHty of quantities ; 
mathematicians have themselves used it to indicate 
equivalence of operations. The force of analogy has been 
so great that writers in most other branches of science 
have employed the same sign. The philologist uses it to 
indicate the equivalence of meaning of words : chemists 
adopt It to signify identity in kind and equality in weight 
of the elements which form two different compounds 
Not a few logicians, for instance Lambert, Drobitsch 
George Bentham,^ Boole,' have employed it as the copula 
of propositions. De Morgan declined to use it for this 
purpose, but still further extended its meaning so as to 
include the equivalence of a proposition with the premises 
from which it can be inferred ; * and Herbert Spencer has 
applied it m a like manner.*^ 

Many persons may think that the choice of a symbol is 
a matter of slight importance or of mere convenience : but 
1 hold that the common use of this sign = in so many 
different meanings is reaUy founded upon a generalisation 
ot the widest character and of the greatest importance- 
one indeed which it is a principal pui-pose of this work to 
explain. The employment of the same sign in different 
cases would be unphilosophical unless there were some real 
analogy between its diverse meanings. If such analogy 
exists, It is not only aUowable, but highly desirable aSd 
even imperative, to use the symbol of equivalence with a 
generahty of meaning corresponding to the generality of 
the principles involved. Accordingly De Morgan's refusal 
to use the symbol in logical propositions indicated his 
opinion that there was a want of analogy between logical 
propositions and mathematical equations. I use the sign 
because I hold the contrary opinion. 

J flam's Literature of Europe, First Ed., vol. ii. p. 444. 

OuUtne of a New SysUm of Logic, London, 1827, ppTi, &. 
; An Investtgation of tJu Laws of Thought,\p. 27, &i.^^' 

aJsZt f/T' PPV^^.'°^- .1° ^18 later wo?k, The Syllabus of a 
a^^jsUm of Logic, he discontinued the use of the sign ^ 

Principles of Ptychology, Second Ed., vol. il pp. 5^*55 




m '' 






I conceive that the sign =■ as commonly employed, always 
denotes some form or degree of sameness, and the particular 
form is usually indicated by the nature of the terms joined 
by it. Thus " 6,720 pounds = 3 tons " is evidently an 
equation of quantities. The formula — X — = + ex- 
presses the equivalence of operations. " Exogens = Dico- 
tyledons " is a logical identity expressing a profound truth 
concerning the character and origin of a most important 
group of plants. 

We have great need in logic of a distinct sign for the 
copula, because the little verb is (or are), hitherto used 
both in logic and ordinary discourse, is thoroughly am- 
biguous. It sometimes denotes identity, as in " St. Paul's 
is the chef-d'osuvre of Sir Christopher Wren ; " but it 
more commonly indicates inclusion of class within class, 
or partial identity, as in " Bishops are members of the 
House of Lords." This latter relation involves identity, 
but requires careful discrimination from simple identity, as 
will be shown further on. 

When with this sign of equality we join two nouns or 
logical terms, as in 

Hydrogen = The least dense element, 
we signify that the object or group of objects denoted by 
one term is identical with that denoted by the other, in 
everything except the names. The general formula 

A = B 
must be taken to mean that A and B are symbols for the 
same object or group of objects. This identity may some- 
times arise from the mere imposition of names, but it may 
also arise from the deepest laws of the constitution of 
nature ; as when we say 

Gravitating matter = Matter possessing inertia, 
Exogenous plants = Dicotyledonous plants, 
Plagihedral quartz crystals = Quartz crystals causing 
the plane of polarisation of light to rotate. 
We shall need carefully to distinguish between relations 
of terms which can be modified at our own will and those 
which are fixed as expressing the laws of nature ; but at 
present we are considering only the mode of expression 
which may be the same in either case. 

Sometimes, but much less frequently, we require a 
symbol to indicate difference or the absence of complete 




sameness. For this purpose we may generalise in like 
manner the symbol -', which was introduced by Wallis 
to signify difference between quantities. The general 

B - C 

denotes that B and C are the names of two objects or 
groups whicli are not identical with each other. Thus 
we may say 

Acrogens ^ Flowering plants. 

Snowdon ^ The highest mountain in Great Britain. 
I shall also occasionally use the sign cos to signify in the 
most general manner the existence of any relation between 
the two terms connected by it. Thus c//i might mean not 
only the relations of equality or inequality, sameness or 
difference, but any special relation of time, place, size, 
causation, &c. in which one thing may stand to another. 
By A C05 B I mean, then, any two objects of thougl^ 
related to each other in any conceivable manner. 

General Formula of Logical Inference, 

The one supreme rule of inference consists, as I have 
said, in the direction to affirm of anything whatever is 
known of its like, equal or equivalent. The SiibstUution 
of Similars is a phrase which seems aptly to express the 
capacity of nmtual replacement existing in any two objects 
wJiich are like or equivalent to a sufficient degree. It is 
matter for furtlier investigation to ascertain when and for 
what purposes a degree of similarity less than complete 
identity is sufficient to warrant substitution. For the 
present we think only of the exact sameness expressed in 
the form 


Now if we take the letter to denote any third con- 
ceivable object, and use the sign c^ in its stated meaninc 
oi tndefimte relation, then the general formula of all 
inlerence may be thus exhibited :— 

^rom A = B :<>: 

we may infer A g6» C 

or, m words— /w whatever relation a thing stands to a 
second thin^, in the same relation it stands to the like or 
eqmvalent of that second thing. The identity between A 







and B allows us indififerently to place A where B was, or 
B where A was ; and there is no limit to the variety of 
special meanings which we can bestow upon the signs 
used in this fonnula consistently with its truth. Thus if 
we first specify only the meaning of the sign coj, we may 
say that if C is tlie weight of B, then G is also the weight 
of A. Similarly 

If C is the father of B, C is the father of A ; 

If C is a fragment of B, C is a fragment of A ; 

If C is a quality of B, C is a quality of A ; 

If C is a species of B, C is a species of A ; 

If C is the equal of B, C is the equal of A ; 
and so on ad infinitum. 

We may also endow with special meanings tlio letter- 
terms A, B, and C, and the process of inference will never 
be false. Thus let the sign ooo mean " is height of," and let 
A = Snowdon, 

B = Highest mountain in England or Wales, 
C = 3.590 feet; 
then it obviously follows since " 3,590 feet is the Height 
of Snowdon," and " Snowdon = the highest mountain iu 
England or Wales," that, " 3,590 feet is the height of the 
highest mountain in England or Wales." 

One result of this general process of inference is that we 
may in any aggregate or complex whole replace any part 
by its equivalent without altering the whole. To alter is 
to make a difference ; but if iu replacing a part I make no 
difference, there is no alteration of the whole. Many 
inferences which have been very imperfectly included in 
logical formulas at once follow. I remember the late Prof. 
De Moi-gan remarking that all Aristotle's logic could not 
prove that " Because a horse is an animal, the head of a 
horse is the head of an animal." I conceive that thia 
amounts merely to replacing in the complete notion head of 
a horse, the term " horse," by its equivalent some animal or 
an animal. Similarly, since 

The Lord Chancellor = The Speaker of the House of 
it follows that 

The death of the Lord Chancellor « The death of the 
Speaker of the House of Lords ; 
and any event, circumstance or thing, which stands iu a 

certain relation to the one will stand in like lelation to the 
other. Milton reasons in this way when he says, in his 
Areopagitica, " Who kills a man, kills a reasonable creature, 
God's image." If we may suppose him to mean 

G^d's image = man = some reasonable creature, 
it follows that " The killer of a man is the killer of some 
reasonable creature," and also " The killer of God's image.'- 
This replacement of equivalents may be repeated over 
and over again to any extent Thus if person is identical 
in meaning with individual, it follows that 

Meeting of persons = meeting of individuals ; 
and if assemblage = meeting, we may make a new 'replace- 
ment and show that 

Meeting of persons = assemblage of individuals. 
We may in fact found upon this principle of substitution 

a most genei-al axiom in the following terms ^ ; 

Same parts samely related mdce same wholes. 
If, for instance, exactly similar bricks and other 
materials be used to build two houses, and they be simi- 
larly placed in each house, the two houses must be similar. 
There are millions of cells in a human body, but if each 
cell of one person were represented by an exactly similar 
cell similarly placed in another body, the two persons 
would be undistinguishable, and would be only numerically 
different. It is upon this principle, as we shall see, that 
all accurate processes of measurement depend. If for a 
weight in a scale of a balance we substitute another 
weight, and the equilibrium remains entirely unchanged 
then the weights must be exactly equal. The general test 
of equality is substitution. Objects are equally bright 
when on replacing one by the other the eye perceives no 
difference. Objects are equal in dimensions .when tested 
by the same gauge they fit in the same manner. Generally 
speaking, two objects are alike so far as when substituted 
one for another no alteration is produced, and vice versd 
when alike no alteration is produced by the substitution. 

» Purt Tjogk, or the Logic of Quality^ p. 14. 

c 2 




The Propagating Power of Similarity, 

The relation of similarity in all its degrees is reciprocal 
So far as things are alike, either roay be substituted for the 
other; and this may perhaps be considered the very 
meaning of the relation. But it is well worth notice that 
there is in similarity a peculiar power of extending itself 
among all the things which are similar. To render a 
number of things similar to each other we need only 
render them similar to one standard object Each coin 
struck from a pair of dies not only resembles the matrix 
or original pattern from wliich the dies were struck, but 
resembles every other coin manufactured from the same 
original pattern. Among a million such coins there are 
not less than 499>999> 5 00,000 pairs of coins resembling 
each other. Similars to the same are similars to all. it 
is one great advantage of printing that all copies of a 
iocument struck from the same type are necessarily 
identical each with each, and whatever is true of one copy 
will be true of every copy. Similarly, if fifty rows of 
pipes in an organ be tuned in perfect unison with one row, 
usually the Principal, they must be in unison with each 
other. Similarity can also reproduce or propagate itself 
ad infinitum : for if a number of tuning-forks be adjusted 
in perfect unison with one standard fork, all instruments 
tuned to any one fork will agree with any instrument 
tuned to any other fork. Standard measures of length, 
capacity, weight, or any other measurable quality, are 
propagated in the same manner. So far as copies of the 
original standard, or copies of copies, or copies again of 
those copies, are accurately executed, they must all agree 
each with every other. 

It is the capability of mutual substitution which gives 
such great value to the modern methods of mechanical 
construction, according to which all the parts of a machine 
are exact facsimiles of a fixed pattern. The rifles used in 
the British army are consti-ucted on the American inter- 
changeable system, so that any part of any rifle can be 
substituted for the same part of another. A bullet fitting 
one rifle will fit all others of the same bore. Sir J. 




Whitworth has extended the same system to the screws 
and screw-bolts used m connecting together the parts of 
machmes, by estabbshing a series of standard screws. 

Anticipations of tJie Fnrmple of Substitution. 

In such a subject as logic it is hardly possible to put 
forth any opinions which have not been in some decree 
previously entertained. The germ at lea^t of every 
doctrine will be found m earHer wi-itei^, and novelty mus^ 
arise chiefly in the mode of harmonising and develop 
deas When I first employed the process and name 0I 
substiution m logic,^ I was led to do so from analogy with 
the familiar mathematical process of substituting for a 
symbol it« value as given in an equation. In writhig my 
hrst logical essay I had a most imperfect conception of the 

^KhTwe^'n7^^Y '' ''^ P^^^^^«' -^ ' ^^^^^l 
as It they were of equal importance, a number of other 

laws which now seem to be but particular cases of the one 
general rule of substitution. 

My second essay, '^The Substitution of Similars " was 
written shortly after I had become aware o™' Jeat 
simplihcation which may be effected by a proper aS 
cation of the principle of substitution/ I w^not Then 
RT.r^^ ""'^ '^^ ^^^^ ^^^' ^h« C^erma^ Wian 

l7Ll"ir^V.'^^^^^^ ^f substitutiorand 

had used the word itself in forming a theory of* the 

syllogism. My imperfect acquaintance with the German 

pSnfT?L ^^?r^^'' but there is no doubt that 

othcr?oS.r '" "^^' '^ '"^^°- '^^^ ^^> ^^d probably 
other logicians, were m some degree famiUar with 
the principle * Even Aristotle's dictum manrreirarded 

modifv ^hif^- I ^^'^ P^'""^^ ^^^' w« have only to 

bn of fvf ^'^'''^. '''. ^^^^^da^ce with the quantifica^ 

tion of the predicate m order to arrive at the^ complete 

' > 








process of substitution.^ The Port-Royal logicians appear 
to have entertained nearly equivalent views, for they 
considered that all moods of the syllogism might bo 
reduced under one general principle.^ Of two premises 
they regard one as the containing proposition (propositio 
continens), and the other as the applicative proposition. 
The latter proposition must always be affirmative, and 
represents that by which a substitution is made; the 
former may or may not be negative, and is that in 
which a substitution is effected. They also show that 
this method will embrace certain cases of complex reason- 
ing which had no place in the Aristotelian syllogism. 
Their views probably constitute the greatest improvement 
in logical doctrine made up to that time since the days 
of Aristotle. But a true reform in logic must consist, 
not in explaining the syllogism in one way or another, 
but in doing away with all the narrow i-estrictions of 
the Aristotelian system, and in showing that there exists 
an infinite variety of logical arguments immediately 
deducible from the principle of substitution of which the 
ancient syllogism forms but a small and not even the 
most important part 

The Logic of Relatives. 

There is a difficult and important branch of logic 
which may be called the Logic of Relatives. If I argue, 
for instance, that because l)aniel Bernoulli was the son 
of John, and John the brother of James, therefore Daniel 
was the nephew of James, it is not possible to prove 
ttiis conclusion by any simple logical process. We re- 
quire at any rate to assume that the son of a brother is 
a nephew. A simple logical relation is that which exists 
between properties and circumstances of the same object 
or class. But objects and classes of objects may also be 
related according to all the properties of time and space. 
I believe it may be shown, indeed, that where an inference 
concerning such relations is drawn, a process of sub- 
stitution is really employed and an identity must exist ; 

* Svhstitution of Similars (1869), p. 9. 

• Port-Royal Logic, traDsl. by Spencer Baynes, pp. 212-219. 
Part III. chap. x. and xi 


but I will not undei-take to prove the assertion in thia 
work. The relations of time and space are logical 
relations of a complicated character demanding much 
abstract and difficult investigation. The subject has been 
treated with such great ability by Peirce,^ De Morgan,* 
Ellis,^ and Harley, that I will not in the present work 
attempt any review of their writings, but merely refer 
the reader to the publications in which they are to be 

» Description of a Notation for the Logic of Relatives, resulting 
from an Am.phJicatton of the Conceptions of Boole's Calculus of Lofjic. 
By C. S. Peirce. Memoirs of the American Academy , vol. ix. Cam- 
bridge, U.S., 1870. 

2 On the Syllogism No IV., and on tiie Logic of Relations. By 
Augustus De Morgan. Transactions of ths Cambridge FhUosophicaX 
Society, vol. x. part ii., i860. 

3 Observations on Boole's Laws of Thought. By the Jate E. Leslie 
Ellis ; conimunicuted by the Rev. Robert Harley, F.R.S. Report of 
Vie British Association, 1870. Report of Sections, p. 12. Alio, On 
Books Laws of Thought. By the Rev. Robert Harley, F.R.S.. ibid. 
p. 14. J» f 



, \ 







»nf, T*r?^'''°°."P'T^ "»« resemblance or differ- 
ence of the things denoted by its tenns. As inference 
treats of the relation between two or more proposE s^ 
a proposition expresses a relation between two or more 
terms. In the portion of this work which t^te of 
Jeduction It will be convenient to follow the usu^^rder 
of exposition. We will consider in succession the va^Sus 
kinds of terms, propositions, and amuments and weTm 
meiice in this chapter with terms 

The simplest and most palpable meaning which can 
belong to a term consiste of s^me. single material obj^t 
such a^ Westminster Abbey. Stonehenge, the Sun. sS 
&c It IS probable that in early stagel of intellect 3 
concrete and palpable things are^ th^ob/ecte S^Sughf 

h«^i«ry); ^"^ '^? 'ew'goise l>is master ammig a 

hundred other persons, and animals of much lower intel 
hgence kpow and discriminate their haunts iTdl such 
acts there is judgment concerning the likeness of XyS 
objecte, but there is little or no"" power of analysfnS 
object and regarding it as a group ^f qualities. ^ ^ 

The dignity of intellect begins with the power of 
separating points of agreement from those of dCncf 
Comparison of two objects may lead us to perceivrth^t 
they are at once like and unlike^ TwoTragmentTof ^ 
may differ entirely in outward form, yet tKay have^e 
same colour, hardness, and texture. Rowere^ych^^ 
m colour may differ in odour. The mind K to r^ 

each obj«5t as an aggregate of qualities, and acquires the 
power of dwelling at will upon one or otheTof those 
^.rl'^T *^ .e^lus'on of the rest. Logical abstraction. 
m short, comes into play, and the mind becomes capable of 
reasoning, not merely about objects which are physically 
comp lete and concrete, but about things which may be 
thought ol sepai-ately in the mind though they exist not 
separately m nature We can think of "the hardness of 

L,^5f' °\-'' ?''''"' "f a flower, and thus produce 
abstract notions, denoted by abstract terms, which will 
form a subject for further consideration. 

nitl w°*® *""° *"*" S^'ieral notions and classes of- 
objects. We cannot fail to observe that the quality hard- 
"f* exists ,n many objects, for instance in many fr^m^nte 

clasTw 1*"^ n-g. the^e together, we^create The 
class Mrrf olyect, which will include, not only the actual 

sli^s i.!^"!!' V^^y ^* "^^^ ^^ «'^«'- ^ «ur 

s3wTrnn^.°^''''T, «P»rt to us aU the contents of 
oCts whXr!^ UBuaUy set any limits to the number of 

we wS„ iT^ ^^.u'""^ *"y ^'"'^ <=1^- At this point 
whi, jfl to perceive the power and generality of thought 

or i Tnfii^f ,"' "* " ''"Sle act to treat of'^indefinitely 
m' ! u \"'^°''*.^y numerous objects. We can safely assert 
eSlt'^"' « t'"e «f any 'one object coming Id^r a 
ctass IS true of any of the other objects so far as thev 

Tcl^''' W^'^'l"''^f '""P^^d L therbSonSnS 
^ve^Xren]l^?''K r' ^Y? * ^^^S in a class uidess 
cla^^n^^^ 1^ }^^^y^ "^ ^' ^^ that is believed of the 

CO .Tideratirj^ ', ^h' ',' ""T^' " ""^'^"^ °f i«°PO^t-nt 
wiisiueration to decide how far and in what manner we 

can safely undertake thus to assign the place of StsTn 
WyTirr" °' '"^'^^ -^-^ constUuCh: 

Two/old Meaning of Oeneral Names. 

are^cST?J!f the "^nin^ of a name is that which we 
e™p«i "^^ °^ *h^° "'« "a^e is used. Now every 

thLkof «.^'''"8"'8toaclass; ^* ""^^ also cause us to 
'"".k of the common qualities possessed by those objects 








fi I 

A name is said to denote the object of tlioiiglit to which it 
may be applied ; it implies at the same time the possession 
of certain qualities or circumstances. The objects denoted 
form the extent of meaning of the term ; the qualities 
implied form the intent of meaning. Crystal is the name 
of any substance of which the molecules are arranged in 
a regular geometrical manner. The substances or objects 
in question form the extent of meaning ; the circumstance 
of having the molecules so arranged forms the intent of 


When we compare general terms together, it may often 
be found that the meaning of one is included in the mean- 
ing of another. Thus all crystals ai*e included among 
material substances^ and all opaque cin/stals are included 
among crystals; here the inclusion is in extension. We 
may also have inclusion of meaning in regard to intension. 
For, as all crystals are material substances, the qualities 
implied by the term material substance must be among 
those implied by crystal. Again, it is obvious that while 
in extension of meaning opaque crystals are but a pai*t of 
crystals, in intension of meaning crystal is but part of 
opaque crystal. We increase the intent of meaning of a 
term by joining to it adjectives, or phrases equivalent to 
adjectives, and the removal of such adjectives of course 
decreases the intensive meaning. Now, concerning such 
changes of meaning, the following all-important law holds 
universally true : — When the intent of meaning of a teim is 
increased the extent is decreased ; and vice versa, when the 
extent is increased the intent is decreased. In short, as one is 
increased the other is decreased. 

This law refers only to logical changes. The number of 
steam-engines in the world may be undergoing a rapid 
increase without the intensive meaning of the name being 
altered. The law will only be verified, again, when there 
is a real change in the intensive meaning, and an adjective 
may often be joined to a noun without making a change. 
Elementary metal is identical with metal; mortal man 
with man; it being a property of all metals to be elements, 
and of all men to be mortals. 

There is no limit to the amount of meaning which a 
term may have. A term may denote one object, or many, 
or an infinite number ' it mav im^ly a single quality, if such 

there be, or a group of any number of qualities, and vet 
the law connecting the extension and intension will in- 
faUibly apply. Taking the general name planet, we 
increase its intension and decrease its extension bv 
prefixing the adjective exteHcrr ; and if we further add 
nearest to tlu earth, there remains but one planet. Mars to 
which the name can then be applied. Singular terms 
which denote a single individual only, come under the 
same law of meaning as general names. They may be 
regarded as general names of which the meaning in exten- 
sion is reduced to a minimum. Logicians have erroneously 
asserted, as it seems to me, that singular t^rms are devoid 
of meaning m intension, the fact being that they exceed 
all other terms m that kind of meaning, as I have else- 
where tned to show.i 

Abstract Terms. 

Comparison of objects, and analysis of the complex 
resemblances and differences which they present, lead us 
to the conception of ahdract qimlUies. We learn to think 
of one object as not only different from another, but as 
dittering m some particular point, such as colour, or 
weight, or size. We may then convert points of agreement 
or difference into separate objects of thought which we 
call qualities and denote by abstract terms. Thus the terra .o 
rediuss means something in which a number of objects 
agree as to colour, and in virtue of which they are caUed 
red. Kedness forms, in fact, the intensive meanins of the 
term red. * 

Abstract terms are strongly distinguished from general 
terms by possessing only one kind of meaning; for ^ they 
denote qualities there is nothing which they cannot in 

Sri'"".? ^- ?^ l^J'"*^^" " ^ " i« ^^'^ ^^^^ of red 
Objects, but It implies the possession by them of the quality 

Sel afsoT S^^?!fr^^/^T T ^^' ??' ^'"^I P^re Logic, p. 6. 

Sheaden^^ llf / ^f r"^ ^^ f '^'i' ^^^'^ ^' ^^P' "' sections and 
giieddens Elements of Logic, London, 1864, pp 14. &c Profpssr.r 

Robertson objects {Mind,\o\, i, p. 2ii) thatPco^se^uLf ^nd 
pro^^ names ; if so it « because I hold that the same i-emf rks app^y 
to proi^r names, which do not seem to me to differ lo^caUv fiSm 
•uif^iar names. 






redness ; but this latter term has one single meaning— the 
quality alone. Thus it arises that abstract terms are in- 
capadle of plurality. Eed objects are numerically distinct 
each from each, and there are multitudes of such objects • 
but redness is a single quality which runs through all 
those objects, and is the same in one as it is in another 
It IS true that we may speak of rednesses, meaning different 
kmds or tints of redness, just as we may speak of colours. 
meaning different kinds of colours. But in distinguLshinJ 
kinds, degrees, or other differences, we render the terms so 
tar concrete. In that they are merely red there is but a 
single nature m red objects, and so far as things are merely 
coloured, colour is a single indivisible quality. Redness, 
so far as it is redness merely, is one and the same every- 
where, and possesses absolute oneness. In virtue of this 
unity we acqiure the power of treating all instances of 
such quabty as we may treat any one. We possess in 
short, genei-al knowledga 

Substantial Terms, 

Logicians appear to have taken little notice of a class of 
terms which partake in certain respects of the character of 
abstract terms and yet are undoubtedly the names of con- 
crete existing things. These terms are the names of 
substances, such as gold, carbonate of lime, nitrogen &c 
We cannot speak of two golds, twenty carbonates of lime' 
or a hundred nitrogens. There is no such distinction 
between the parts of a uniform substance as will allow of 
a discrimination of numerous individuals. The qualities of 
colour, lustre, malleability, density, &c., by which we 
recogmse gold, extend through its substance irrespective of 
particular size or shape. So far as a substance is gold it 
is one and the same everywhere ; so that terms of this 
kind, which I propose to caU substantial terms, possess 
the peculiar unity of abstract terms. Yet they are not 
abstract; for gold is of course a tangible visible body 
^tirely concrete, and existing independently of othe^ 

It is only when, by actual mechanical division, we break 
up the uniform whole which forms the meaning of a 
substantial term, that we introduce number. Piece of gold 




is a term capable of plurality ; for there may be a great 
many pieces discriminated either by their various shapes 
and sizes, or, in the absence of such marks, by simul- 
taneously occupying different parts of space. In substance 
they are one ; as regards the properties of space they are 
many.i We need not further pursue this question, which 
involves the distinction between unity and plurality, until 
we consider the principles of number in a subsequent 

Collective Terms. 

We must clearly distinguish between the collective and 
the general meanings of terms. The same name may be 
used to denote the whole body of existing objects of a 
certain kind, or any one of those objects taken separately. 
" Man " may mean the aggregate of existing men, which we 
sometimes describe as mankind; it is also the general 
name applying to any man. The vegetable kingdom is 
the name of the whole aggregate of plants, but " plant " 
itself is a general name applying to any one or other plant. 
Every material object may be conceived as divisible into 
parts, and is therefore collective as regards those parts. 
Tlie animal body is made up of cells and fibres, a crystal 
of molecules; wherever physical division, or as it has been 
called partition, is possible, there we deal in reality with a 
collective whola Thus the greater number of general 
terms are at the same time collective as regards each 
individual whole which they denote. 

It need hardly be pointed out that we must not infer of 
a collective whole what we know only of the parts, nor of 
the parts what we know only of the whole. The relation 
of whole and part is not one of identity, and does not 
allow of substitution. There may nevertheless be qualities 
which are true alike of the whole and of its parts. A 
number of organ-pipes tuned in unison produce an aggre- 
gate of sound which is of exactly the same pitch as each 

* Professor Robertson has criticised my introduction of "Substantial 
l.erms {Mind, vol. i. p. 210), and objects, perhaps correctly, that the 
aistinction if valid is extra-logical. I am inclined to think, however, 
uiat the doctrine of terms ia, strictly speaking, for the most part 



W- !i ' i I i 



separate. sound. In the case of substantial terms, certain 
qualities may be present equally in each minutest part as 
in the whole. The chemical nature of the largest mass of 
pure carbonate of lime is the same as the nature of the 
smallest particle. In the case of abstract terms, again, we 
cannot di-aw a distinction between whole and part ; what 
ia true of redness in any case is always true of redness, so 
far as it is merely red. ^ 

Synthesis of TenM, 

We continually combine simple terms together so as to 
form new terms of more complex meaning. Thus, to 
increase the intension of meaning of a term we write it 
with an adjective or a phrase of adjectival nature. By 
joining "brittle" to "metal," we obtain a combined term, 
"brittle metal," which denotes a certain portion of the 
metals, namely, such as are selected on account of pos- 
sessing the quality of brittleness. As we have already 
seen, " brittle metal " possesses less extension and greater 
intension than metal. Nouns, prepositional phrases, parti- 
cipial phrases and subordinate propositions may also be 
added to terms so as to increase their intension and 
decrease their extension. 

In our symbolic language we need some mode of indi- 
cating this junction of terms, and the most convenient 
device will be the juxtaposition of the letter-terms. Thus 
if A mean brittle, and B mean metal, then AB will mean 
brittle metal Nor need there be any limit to the number 
of letters thus joined together, or the complexity of the 
notions which they may represent. 
Thus if we take the letters 

P = metal, 

Q = white, 

R = monovalent, 

S = of specific gravity iO'5, 

T = melting above 1000° C, 

V = good conductor of heat and electricity, 
then we can form a combined term PQRSTV, which will 
denote "a whit« monovalent metal, of specific gravity 10 5, 
melting above 1000° C, and a good conductor of heat and 




There are many grammatical usages concerning tho 
junction of words and phrases to which we need pay no 
attention m logic. We can never say in ordinary languafje 
"of wood table,' meaning "table of wood;" but we may 
consider "of wood" as logically an exact equivalent of 
" wooden ; so that if 

X ~ of wood, 

Y = table, 

there is no reason why, in our symbols, XY should not be 

just as correct an expression for " table of wood " as YX 

In this case indeed we might substitute for "of wood " the 

correspondmg adjective " wooden," but we should often fail 

to find any adjective answering exactly to a phrase. There 

is no single word by which we could express the notion 

'of specific gravity 105 : " but logically we may consider 

these words as forming an adjective; and denoting this by 

b and metal by P, we may say that SP means " metal of 

specific gmvity 105." It is one of many advantages in 

these blank letter-symbols that they enable us completely 

to neglect all grammatical peculiarities and to fix our 

attention solely on the purely logical relations involved 

Investigation will probably show that the rules of grammar 

are mainly founded upon traditional usage and have little 

ogical signification. This indeed is sufficiently proved by 

tlie wide grammatical differences which exist between 

languages, though the logical foundation must be the 


Syniholic Expression of the Law of Contradiction. 

T '^'^^VXS^'^®^^® ^^ ^™^ ^^ subject to the all-important 
law of Thought, described in a previous section (p. c) and 
called the Law of Contradiction. It is self-evident that no 
quality can be both present and absent at the same time 
and place. This fundamental condition of all thought and 
ot aU existence is expressed symbolically by a rule that a 
terni and its negative shall never be allowed to come into 
combination. Such combined terms as Aa, Bb, Cc, &c are 
seii-contradictory and devoid of all inteUigible meaning 
It they could represent anything, it would be what cannot 

l^nU r '^''''^ ^""^'^ ^^ imagined in the mind. They 
can therefore only ent^r into our consideration to suffer 


r i\ 




immediate exclusion. The criterion of false reasoning, as we 
shall find, is that it involves self-contradiction, the affirm- 
ing and denying of the same statement. We might repre- 
sent tne object of all reasoning as the separation of the 
consistent and possible from the inconsistent and impossi- 
ble ; and we cannot make any statement except a truism 
without implying that certain combinations of terms are 
contradictory and excluded from thought. To assert that 
" all A's are B's " is equivalent to the assertion that " A's 
which are not B's cannot exist." 

It will be convenient to have the means of indicating 
the exclusion of the self-contradictory, and we may use the 
familiar sign for nothing, the cipher o. Thus the second 
law of thought may be symbolised in the forms 
Aa = o ABh = o ABCa = o 
We may variously describe the meaning of o in logic as 
the non-existejit, the impossible, the self-incoumtent, the 
inconceivable. Close analogy exists between this meaning 
and its mathematical signification. 

Certain Special Conditions of Logical Symbols. 

In order that we may argue and infer truly we must 
treat our logical symbols according to the fundamental 
laws of Identity and Difference. But in thus using our 
symbols we shall frequently meet with combinations of 
which the meaning will not at first sight be apparent If 
in one case we learn that an object is " yellow and round," 
and in another case that it is " round and yellow," there 
arises the question whether these two descriptions are 
identical in meaning or not. Again, if we proved that an 
object was " round round," the meaning of such an expres- 
sion would be open to doubt. Accordingly we must take 
notice, before proceeding further, of certain special laws 
which govern the combination of logical terms. 

In the first place the combination of a logical term with 
itself is without effect, just as the repetition of a statement 
does not alter the meaning of the statement ; " a round 
round object" is simply "a round object." What is 
yellow yellow is merely yellow; metallic metals cannot 
differ from metals, nor circular circles from circles. In nu/ 





symbolic language we may similarly hold that A A is iden- 
tical with A, or 

A = AA = AAA = &c. 
The late Professor Boole is the only logician in modern 
times who has drawn attention to this remarkable property 
of logical terms ; ^ but in place of the name which he gave 
to the law, I have proposed to call it The Law of Simpli- 
city.* Its high importance will only become apparent 
when we attempt to determine the relations of logical and 
mathematical science. Two symbols of quantity, and only 
two, seem to obey this law ; we may say that i x i = i, 
and 0x0 = (taking o to mean absolute zero or i - i) ,' 
there is apparently no other number which combined with 
itself gives an unclianged result. I shall point out, how- 
ever, in the chapter upon Number, that in realitv all 
numerical symbols obey this logical principle. 

It is curious that this Law of Simplicity, though almost 
unnoticed in modern times, was known to Boethius, who 
makes . a singular remark in his treatise De Trinit'ate et 
Unitate Dei (p. 959). He says : *• If I should say sun, 
sun, sun, I should not have made three suns, but I should 
have named one sun so many times." » Ancient discussions 
about the doctrine of the Trinity drew more attention 
to subtle questions concerning the nature of unity and 
plui-ality than has ever since been given to them. 

It is a second law of logical symbols that order of com- 
bmation is a matter of indifference. " Rich and rare gems " 
are the same as " rare and rich gems," or even as " gems, 
rich and rare." Grammatical, rhetorical, or poetic usage 
may give considerable significance to order of expression. 
The limited power of our minds prevents our grasping 
many ideas at once, and thus the order of statement may 
produce some effect, but not in a simply logical manner. 
All life proceeds in the succession of time, and we are 
obliged to write, speak, or even think of things and their 
qualities one after the other ; but between the things and 
tneir qualities there need be no such relation of order in 

T^^^^^f^*^^ ^wa/ym 0/ Logic, Cambridge, 1847, p. 17. An 
Jnte^atton of the Latoa of Thought, London, iSsi p. 31. ^ ^ 
^ rure Logic, p. 15. 

totie«'p^icale^' ^^' ^^ ^''^' ''''° ^ "^^"^ ^^'''""'' *''^ ^''^ 








'J iH' 

1 ■ I 


time or space. The sweetness of sugar is neither before 
nor after its weight and solubility. Tlie hardness of a 
metal, its colour, weight, opacity, malleability, electric and 
chemical properties, are all coexistent and coextensive, per- 
vading the metal and every part of it in perfect community, 
none before nor after the others. In our words and symbols 
we cannot observe this natural condition ; we must name 
one quality first and another second, just as some one must 
be the first to sign a petition, or to walk foremost in a pro- 
cession. In nature there is no such precedence. 

I find that the opinion here stated, to the effect that 
relations of space and time do not apply to many of our 
ideas, is clearly adopted by Hume in his celebrated Trea- 
tise on Human Nature (vol. i. p^ 410). He says :* — " An 
object may be said to be no where, when its parts are not so 
situated with respect to each other, as to form any figure 
or quantity ; nor the whole with respect to other bodies so 
as to answer to our notions of contiguity or distance. Now 
this is evidently the case with all our perceptions and 
objects, except those of sight and feeling. A moral reflection 
cannot be placed on the right hand or on the left hand 
of a passion, nor can a smell or sound be either of a circular 
or a square figure. These objects and perceptions, so far 
from requiring any particular place, are absolutely incom- 
patible with it, and even the imagination cannot attribute 
it to them." 

A little reflection will show that knowledge in the 
highest perfection would consist in the simultamaus pos- 
session of a multitude of facts. To comprehend a 
science perfectly we should have every fact present with 
every other fact. We must write a book and we must read 
it successively word by word, but how infinitely higher 
would be our powers of thought if we could grasp the 
whole in one collective act of consciousness ! Compared 
with the brutes we do possess some slight approximation 
to such power, and it is conceivable that in the indefinite 
future mind may acquire an increase of capacity, and be 
less restricted to the piecemeal examination of a subject. 
Bat I wish here to make plain that there is no logical 
foundation for the successive character of thought and 
reasoning unavoidable under our present mental conditions. 

' Book i., Part it., Section 8. 

We are logically weak and imperfect in reject of tlve fact 
thai we are obliged to think of one thing after another. We 
must describe metal as " hard and opaque," or " opaque and 
hard," but in the metal itself there is no such difference of 
order ; the properties are simultaneous and coextensive in 

Setting aside all grammatical peculiarities which render 
a substantive less moveable than an adjective, and dis- 
regarding any meaning indicated by emphasis or marked 
order of words, we may state, as a general law of logic, 
that AB is identical with BA, or AB = BA. Similarly, 
ABC = ACB = BCA = &c. 

Boole first drew attention in recent years to this pro- 
perty of logical terms, and he called it the property of 
Commutativeness.^ He not only stated the law with the 
utmost clearness, but pointed out that it is a Law of 
Thought rather than a Law of Things. I shall have in 
various parts of this work to show how the necessary im- 
perfection of our symbols expressed in this law clings to 
our modes of expression, and introduces complication into 
the whole body of mathematical formulae, which are really 
founded on a logical basis. 

It is of course apparent that the power of commutation 
belongs only to terms related in the simple logical mode of 
synthesis. No one can confuse " a house of bricks" with 
" bricks of a house," " twelve square feet " with " twelve feet 
square," "the water of crystallization" with '* the crystalliza- 
tion of water." All relations which involve differences of time 
and space are inconvertible ; the higher must not be made to • 
change places with the lower, nor the first with the last. For 
the parties concerned there is all the difference in the world 
between A killing B and B kiUing A. The law of com- 
mutativeness simply asserts that difference of order does 
not attech to the connection between the properties and 
circumstances of a thing— to what I call simple logical 

fj ^^if^/ '^^^*^^^y P- 2Q. It is pointed out in the preface to this 
Second Edition that Leibnitz was acquainted with the Laws oS 
Simplicity and of Commutativeneaa, 

r 1 




! >i 





We now proceed to consider the variety of fonns of pro- 
positions in which the truths of science must be expressed. 
I shall endeavour to show that, however diverse these 
forms may be, they all admit the application of the one 
same principle of inference that what is true of a thing is 
true of the like or same. This principle holds true what- 
ever be the kind or manner of the likeness, provided 
proper regard be had to its nature. Propositions may 
assert an identity of time, space, manner, quantity, degree, 
or any other circumstance in which things may agree or 

We find an instance of a proposition concerning time in 
the following : — " The year in which Newton was born, 
was the year in which Galileo died." This proposition 
expresses an approximate identity of time between two 
events; hence whatever is true of the year in which 
Galileo died is true of that in which Newton was born, 
and vice versd, " Tower Hill is the place where Raleigh 
was executed " expresses an identity of place ; and what- 
ever is true of the one spot is true of the spot otherwise 
defined, but in reality the same. In ordinary language we 
have many propositions obscurely expressing identities 
of number, quantity, or degree. " So many men, so many 
minds," is a proposition concerning number, that is to say, 
an equation; whatever is true of the number of men is 
true of the number of -minds, and vice versd. " The density 
of Mars is (nearly) the same as that of the Earth," " The 
force of gravity is directly as the product of the masses, and 

inversely as the square of the distance," are propositions 
concerning magnitude or degree. Logicians have not paid 
adequate attention to the great variety of propositions 
which can be stated by the use of the little conjunction 
as, together with so. " As the home so the people," is a 
proposition expressing identity of manner; and a great 
number of similar propositions all indicating some kind of 
resemblance might be quoted. Whatever be the special 
kind of identity, all such expressions are subject to the 
great principle of inference ; but as we shall in later 
parts of this work treat more particularly of inference in 
cases of number and magnitude, we will here confine our 
attention to logical propositions which involve only notions 
of quality. 

Simple IdeiUities, 

The most important class of propositions consists ofc 
those which fall under the formula \ 

A = B, 
and may be called simple identities. I may instance, in 
the first place, those most elementary propositions which 
express the exact similarity of a quality encountered in 
two or more objects. I may compare the colour of the 
Pacific Ocean with that of the Atlantic, and declare them 
identical. I may assert that '* the smell of a rotten ^gg is 
like that of hydrogen sulphide ; " " the taste of silver hypo- 
sulphite is like that of cane sugar ; " " the sound of an 
earthquake resembles that of distant artillery." Such are 
propositions stating, accurately or otherwise, the identity 
of simple physical sensations. Judgments of this kind 
are necessarily pre-supposed in more complex, judgments. 
If I declare that " this coin is made of gold," I must base 
the judgment upon the exact likeness of the substance in 
several qualities to other pieces of substance which are 
undoubtedly gold. I must make judgments of the colour, 
the specific gravity, the hardness, and of other mechanicaJ 
and chemical properties ; each of these judgments is ex 
pressed in an elementary proposition, " the colour of this 
coin is the colour of gold," and so on. Even when we 
establish the ide'itity of a thing with itself under a 
dilferent name or aspect, it i* by distinct judgments 



ilif (; ' 

j\| ^ f ! 






concerning single circumstances. To prove that the 
Homeric x"'^^^^ is copper we must show the identity of 
each quality rticorded of ^oXko^ with a quality of copper. 
To establish Deal as the landing-place of Caesar, all material 
circumstances must be shown to agree. If the modern 
Wroxeter is the ancient Uriconium, there must be the like 
agreement of all features of the country not subject to 
alteratio3i by time. 

Such identities must be expressed in the form A = B. 
We may say 

Colour of Pacific Ocean = Colour of Atlantic Ocean. 

Smell of rotten egg = Smell of hydro^^en sulphide. 
In these and similar propositions we assert identity of 
single qualities or causes of sensation. In the same form 
we may also express identity of any gi-oup of qualities, as 

X<i^fco<i = Copper. 

Deal = Landing-place of Caesar. 
A multitude of propositions involving singular terms fall 
into the same form, as in 

The Pole star = The slowest-moving star. 

Jupiter = The greatest of the planets. 

The ringed planet = The planet having seven satel- 

The Queen of England = The Empress of India. 

The number two = The even prime number. 

Honesty = The best policy. 
In mathematical and scientific theories we often meet 
with simple identities capable of expression in the same 
form. Thus in mechanical science " The process for finding 
the resultsmt of forces = the process for finding the re- 
sultant of simultaneous velocities." Theorems in geometry 
often give results in this form, as 

Equilateral triangles = Equiangular triangles. 

Circle = Finite plane curve of constant curvature. 

Circle = Curve of least perimeter. 
The more profound and important laws of nature are 
often expressible in the form of simple identities; in 
addition to some instances which have already been given, 
I may suggest, 

Crystals of cubical system = Crystals not possessing 
the power of double refiactiou. 




All definitions are necessarily of this form, whether the 
objects defined be many, few, or singular. Thus we may say, 

Common salt = Sodium chloride. 

Chlorophyl = Green colouring matter of leaves. 

Square = Equal-sided rectangle. 
It is an extraordinary fact that propositions of this 
elementary form, all-important and very numerous as they 
are, had no recognised place in Aristotle's system of Logic. 
Accordingly their importance was overlooked until verv 
recent times> and logic was the most deformed of sciences. 
But it is impossible that Aristotle or any other person 
should avoid constantly using them ; not a term could be 
defined without their use. In one place at least Aristotle 
actually notices a proposition of the kind. He observes • 
" We sometimes say that that white thing is Socrates, or 
that the object approaching is (/allias."^ Here we certainly 
have simple identity of terms ; but he considered such 
propositions purely accidental, and came to the unfoitunate 
conclusion, that " Singulars cannot be predicated of other 

Propositions may also express the identity of extensive 
groups of objects taken collectively or in one connected 
whole ; as when we say, 

The Queen, I^rds, and Commons = The Legislature of 
the United Kingdom. 
When Blackstone asserts that " The only true and natural 
foundation of society are the wants and fears of individuals," 
we must interpret him as meaning that the whole of the 
wants and fears of individuals in the aggregate form the 
foundation of society. But many propositions which 
might seem to be collective are but groups of singular 
propositions or identities. When we say " Potassium and 
sodium are the metallic bases of potash and soda," we 
obviously mean. 

Potassium = Metallic base of potash ; 

Sodium = Metallic base of soda. 
It is the work of grammatical analysis to separate the 
various propositions often combined into a single sentence 
Logic cannot be properly required to interpret the forms 
and devices of language, but only to treat the meaning 
when clearly exhibited. 

Prior Analytics^ L cap. xxvii. v 










1 1 if 

\ 4 f 



Partial Identities. 

A second highly important kind of proposition is that 
which I propose to call a partial identity. When we say 
that "All mammalia are vertebrata," we do not mean that 
mammalian animals are identical with vertebrate animals, 
but only that the mammalia form a part of the class verte- 
hrata. Such a proposition was regarded in the old logic as 
asserting the inclusion of one class in another, or of an 
object in a class. It was called a universal affirmative pro- 
position, because the attribute vertebrate was affirmed of the 
whole subject mammalia ; but the attribute was said to be 
undistrihUedf because not all vertebrata were of necessity 
involved in the proposition. Aristotle, overlooking the im- 
portance of simple identities, and indeed almost denying 
their existence, unfortunately founded his system upon the 
notion of inclusion in a class, instead of adopting the basis 
of identity. He regarded inference as resting upon the rule 
that what is true of the containing class is true of the 
contained, in place of the vastly more general rule that 
what is true of a class or thing is true of the like. Thus 
he not only reduced logic to a fragment of its proper self, 
but destroyed the deep analogies which bind together 
logical and mathematical reasoning. Hence a crowd of 
defects, difficulties and errors which will long disfigure the 
first and simplest of the sciences. 

It is surely evident that the relation of inclusion rests 
upon the relation of identity. Mammalian animals cannot 
be included among vertebrates unless they be identical with 
part of the vertebrates. Cabinet Ministers are included 
almost always in the class Members of Parliament, because 
they are identical with some who sit in Parliament. We 
may indicate this identity with a part of the larger class in 
various ways ; as for instance, 

Mammalia = part of the vertebrata. 

Diatomace8e = a class of plants. 

Cabinet Ministers = some members of Parliament. 

Iron = a metal. 
In ordinary language the verbs is and are express mere 
inclusion more often than not. Men are mortals, means 


that men form a part of the class mortal ; but great con- 
fusion exists between this sense of the verb and that in 
which it expresses identity, as in " The sun is the centre of 
the planetary system." The introduction of the indefinite 
article a often expresses partiality ; when we say " Iron is 
a metal" we clearly mean that iron is one only of several 

Certain recent logicians have proposed to avoid the 
indefiniteness in question by what is called the Quanti- 
fication of the Predicate, and they have generally used the 
-^little word some to show that only a part of the predicate 
is identical with the subject Some is an indeterminate 
adjedive ; it implies unknown qualities by which we might 
select the part in question if the qualities were known, but 
it gives no hint as to their nature. I might make use of 
such an indeterminate sign to express partial identities in 
this work. Thus, taking the special symbol V = Some, the 
general form of a partial identity would be A = VB, and in 
Boole's Logic expressions of the kind were much used. 
But I believe that indeterminate symbols only introduce 
complexity, and destroy the beauty and simple universality 
of the system which may be created without their use. A 
vague word like some is only used in ordinary language by 
ellipsis, and to avoid the trouble of attaining accuracy.' 
We can always employ more definite expressions if we 
hke; but when once the indefinite some is introduced we 
cannot replace it by the special description. We do not 
know whether »ome colour is red, yellow, blue, or what it 
is ; but on the other hand red colour is certainly some 

Throughout this system of logic I shall dispense with 
such indefinite expressions ; and this can readily be done 
by substituting one of the other terms. To express the 
proposition " All A's are some B's " I shall nou use the form 
A = VB. but 

A = AB. 

This formula states that the class A is identical with the 
class AB ; and as the latter must be a part at least of the 
class B, it implies the inclusion of the class A in that of 
B. We might represent our former example thus. 

Mammalia =s Mammalian vertebrata. 
This proposition asserts identity between a part (or it may 






1 m 




be the whole) of the vertebrata and the mammalia. If it is 
asked What part ? the proposition affords no answer, except 
that it is the part which is mammalian ; but the assertion 
" mammalia = some vertebrata " tells us no mora 

It is quite likely that some readers will think this 
mode of representing the universal affirmative proposition 
artificial and complicated. I will not undertake to con- 
vince them of the opposite at this point of my exposition. 
Justification for it will be found, not so much in the im- 
mediate treatment of this proposition, as in the general 
harmony which it will enable us to disclose between all 
parts of reasoning. I have no doubt that this is the 
critical difficulty in the relation of logical to other forms of 
reasoning. Grant this mode of denoting that " all A's are 
B*s," and I fear no further difficulties ; refuse it, and we find 
want of analogy and endless anomaly in every direction. It 
is on general grounds that I hope to show overwhelming 
reasons for seeking to reduce every kind of proposition to 
the form of an identity. 

I may add that not a few logicians have accepted this 
view of the universal affirmative proposition. Leibnitz, in 
his IHficultates Qucedam Logicce^ adopts it, saying, " Omne 
A est B ; id est {equivalent AB et A, seu A non B est non- 
ens." Boole employed the logical equation x = xy con- 
currently with x = vy; and Spalding ^ distinctly says that 
the proposition " all metals are minerals " might be de- 
scribed as an assertion of partial identity between the two 
classes. Hence the name which I have adopted for the 

^ , Limited Identities, 

An important class ot propositions have the form 

AB = AC, 
expressing the identity of the class AB with the class AC. 
In other words, " Within the sphere of the class A, all the 
B's are all the C's ; " or again, " The B's and C's, which are 
A*s, are identical." But it will be observed that nothiug is 
asserted concerning things which are outside of the class 
A ; and thus the identity is of limited extent. It is the 
proposition B = C limited to the sphere of things called A. 

> Encyclaptzdia Britannicaf Eighth Ed. art. Logic, sect. 37, note. 
8vo reprint, p. 79. 

Thus we may say, with some appi-oximation to truth, that 
" Large plants are plants devoid of locomotive power." 

A barrister may make numbers of most general state- 
ments concerning the relations of persons and things in the 
course of an argument, but it is of course to be understood 
that he speaks only of persons and things under the 
English Iaw. Even mathematicians make statements 
which are not true with absolute generality. They say 
that imaginary roots enter into equations by pairs ; but this 
is only true under the tacit condition that the equations in 
question shall not have imaginary coefficients.^ The uni- 
verse, in short, within which they habitually discourse is 
that of equations with real coefficients. These implied 
limitations form part of that great mass of tacit knowledge 
which accompanies all special arguments. 

To Do Morgan is due the remark, that we do usually 
think and argue in a limited universe or sphere of notions, 
even when it is not expressly stated.* 

It is worthy of inquiry whether all identities are not 
really limited to an implied sphere of meaning. When we 
make such a plain statement as " Gold is malleable " we 
obviously speak of gold only in its solid state ; Avhen we 
say that " Mercury is a liquid metal " we must be under- 
stood to exclude the frozen condition to which it may be 
reduced in the Arctic regions. Even when we take such a 
fundamental law of nature as "All substances gravitate," 
we must mean by substance, material substance, not in- 
cludmg that basis of heat, light, and electrical undulations 
which occupies space and possesses many wonderful me- 
chanical properties, but not gravity. The proposition then 
is really of the form 

Material substance = Material gravitating substance. 

Negative Propositions. 

In every act of intellect we are engaged with a certain 
laentity or difference between things or sensations compared 
vB',- ^^^*^herto I have treated only of identities ; and 
yec It might seem that the relation of difference must be 

t^ph^l Tr^"^*-^ the Root of any Function. Cambridge Philo- 
^nical Transactions, 1867, vol xi. p. 25. ^ 

^Uabui of a propoted SytUm of Logic, §§ 122, 123. 





infinitely more common than that of likeness. One thing 
may resemble a great many other things, but then it diflfers 
from all remaining things in the world. Diversity may 
almost be said to constitute life, being to thought what 
motion is to a river. The perception of an object involves 
its discrimination from all other objects. But we may 
nevertheless be said to detect resemblance as often as we 
detect difference. We cannot, in fact, assert the existence 
of a difference, without at the same time implying the 
existence of an agreement. 

If I compare mercury, for instance, with other metals, 
and decide that it is not solid, here is a difference between 
mercury and solid things, expressed in a negative propo- 
sition ; but there must be implied, at the same time, an 
agreement between mercury and the other substances 
which are not solid. As it is impossible to separate the 
vowels of the alphabet from the consonants without at the 
same time separating the consonants from* the vowels, so I 
cannot select as the object of thought solid things, without 
thereby throwing together into another class all things 
which are not solid. The very fact of not possessing a 
quality, constitutes a new quality which may be the ground 
of judgment and classification. In this point of view, 
agreement and difference are ever the two sides of the same 
act of intellect, and it becomes equally possible to express 
the same judgment in the one or other aspect 

Between atfii-mation and negation there is accordingly a 
perfect equilibrium. Every affirmative proposition implies 
a negative one, and vice versd. It is even a matter of in- 
difference, in a logical point of view, whether a positive or 
negative term be used to denote a given quality and the 
class of things possessing it. If the ordinary state of a 
man's body be called good health, then in other circumstances 
he is said not to he in good health ; but we might equally 
describe him in the latter state as sickly, and in his normal 
condition he would be not sickly. Animal and vegetable 
substances are now called organic, so that the other sub- 
stances, forming an immensely greater part of the globe, are 
described negatively as inorganic. But we might, with at 
least equal logical correctness, have described the prepon- 
derating class of substances as mineral, and then vegetable 
and animal substances would have been non-mineral. 




It is plam that any positive term and its corresponding 
negative divide between them the whole universe of 
thought : whatever does not fall into one must fall into the 
other, by the third fundamental Law of Thought, the Law 
of Duality. It follows at once that there are two modes 
of representing a difference. Supposing that the things 
represented by A and B are found to differ, we may indicate 
(see p. 17) the result of the judgment by the notation 

A *- B. 

We may now represent the same judgment by the assertion 
that A agrees with those things which differ from B or 
that A agrees with the not-B's. Using our notation 'for 
negative terms (see p. 14), we obtain 

as the expression of the ordinary negative proposition. 
Thus if we take A to mean quicksilver, and B solid, then 

we have the following proposition : 

Quicksilver = Quicksilver not-solid 

There may also be several other classes of negative pro- 
positions, of which no notice wa^ taken in the old locric 
We may have cases where all A's are not-B's, and at the 
same time all not-B's are A's; there may, in short, be 
a simple identity between A and not-B, which ma^ be 
expressed m the form ^ 

A , -^ = ^• 

An example of this form would be 

Wn , ^e^f^ctora of electricity = non-electiics. 

duchon ^L ? ^'T^""^^^ l"^^^ ^^ ^^^^ ^ results of de- 
duction, with simple, partial, or limited identities between 
negative terms, as in the forms oeDween 

Tf lA^^^* a = <a), aC = 5C, etc. 

It would be possible to represent affirmative pronositions 
in the negative form. Thus '^ron is solid,'' m^Tbe^^^ 
orTalln'' 7"^^ ^ r ' ^^^«^^^^>" ^^ " IroA is nof fluid » 
tTe&i^b?/^: 'i^ ^^ "-n," and "not-soHd," 

alUroS^^^^^ very strong reasons why we should employ 
prSbvthp«\\^f'.^^'?^^^^ ^^™- ^U inference 
tKpreLd in ?^^^^^ equivalents, and a proposi- 

aU \fTFr^ ^^^ ^^™ 0^ ^ identity is ready to yield 

luiiy shown, we can infer in a negative proposition, 







but not by it. Difference is incapable of becoming tlie 
ground of inference ; it is only the implied agreement with 
other differing objects which admits of deductive reason- 
ing; and it will always be found advantageous to employ 
propositions in the form which exhibits clearly the implied 

Conversion of Propositions. 

The old books of logic contain many rules concerning 
the conversion of propositions, that is, the transposition ol" 
the subject and predicate in such a way as to obtain a new 
proposition which will be true when the original proposi- 
tion is true. The reduction of every proposition to the form 
of an identity renders all such rules and processes needless. 
Identity is essentially reciprocal. If the colour of the 
Atlantic Ocean is the same as that of the Pacific Ocean, 
that of the Pacific must be the same as that of the Atlantic. 
Sodium chloride being identical with common salt, common 
salt must be identical with sodium chlorida If the number 
of windows in Salisbury Cathedral equals the number of 
days in the year, the number of days in the year must 
equal the number of the windows. Lord Chesterfield was 
not wrong when he said, *'I will give anybody their choice 
of these two truths, which amount to the same thing ; He 
who loves himself best is the honestest man; or, The 
honestest man loves himself best" Scotus Erigena exactly 
expresses this reciprocal character of identity in saying, 
"There are not two studies, one of philosophy and the 
other of religion ; true philosophy is true religion, and true 
religion is true philosophy." 

A mathematician would not think it worth while to 
mention that if a: = y then also y = a;. He would not con- 
sider these to be two equations at all, but one equation 
accidentally written in two different manners. In written 
symbols one of two names must come first, and the other 
second, and a like succession must perhaps be observed in 
our thoughts: but in the relation of identity. there is no 
need for succession in order (see p. 33) -, each is simul- 
taneously equal and identical to the other. These remarks 
will hold true both of logical and mathematical identity ; 
6? that I shall consider the two forms 


A = B and B = A ^ 
to express exactly the same identity differently written 
All need for rules of conversion disappears, and there will 
be no single proposition in the system which mav not be 
written with either end foremost. Thus A = AB is the 
same as AB = A, aC = bG is the same as &C = aC and so 
forth. ' 

The same remarks are partially true of differences and 
inequalities, which are also reciprocal to the extent that 
one thing cannot differ from a second without the second 
diffenng from the first. Mars differs in colour from 
Venus, and Venus must differ from Mars. The Earth differs 
from Jupiter m density ; therefore Jupiter must differ from 
the ^rth Speaking generally, if A «- B we shaU also 
have B - A and these two forms may be considered ex- 
pressions of the same difference. But the relation of 
differing thmgs is not wholly reciprocal. The density of 
Jupiter does not differ from that of the Earth in the same 
way that that of the Earth differs from that of Jupiter 
The change of sensation which we experience in parsing 
from Venus to Mars is not the same as what we experience 
m passing bwjk to Venus, but just the opposite in nature. 
The colour of the sky is lighter than that of the ocean ; 
therefore that of the ocean cannot be lighter than that of 
the sky, but darker. In these and all similar cases we gain 
a notion of direction or character of change, and resulte of 
immense importance may be shown to rest on this, notion, 
^or the present we shall be concerned with the mere fact 
ot Identity existing or not existing. 

Two/old Interpretation of Propositions. 

^IV"^^' ^ "^^ ^^""^ ^^^"^ (P- 25), may have a meaning 
either in extension or intension ; and according as one cS 
the other meamng is attributed to the terms of a proposi- 
lion, so may a different interpretation be assicmed to the 
proposition itself. When the Wms are abst^Twe must 
reaa them in intension, and a proposition connecting such 
terms must denote the identity or non-identity of the 

qualities respectively denoted by the terms. Thus if we 

Equality = Identity of magnitude. 



1^ f ( 


. ill. 

the assertion means that the circumstance of being equal 
exactly corresponds with the circumstance of being 
identical in magnitude. Similarly in 

Opacity = Incapability of transmitting light, 
the quality of being incapable of transmitting light is de- 
clared to be the same as the intended meaning of the word 

Wlien general names form the terms of a proposition we 
may apply a double interpretation. Thus 

Exogens = Dicotyledons 
means either that the qualities which belong to all exogens 
are the same as those which belong toall dicotyledons, orelse 
that every individual falling under one name falls equally 
under the other. Hence it may be said that there are two 
distinct fields of logical thought. We may argue either by 
the qualitative meaning of names or by the quantitative, 
that is, the extensive meaning. Every argument in- 
volving concrete pluml terms miglit be converted into 
one involving only abstract singular terms, and vice 
versd. But there are reasons for believing that the 
intensive or qualitative form of reasoning is the primaiy 
and fundamental one. It is sufficient to point out that the 
extensive meaning of a name is a changeable and fleeting 
thing, while the intensive meaning may nevertheless remain 
fixed. Very numerous additions have been lately made 
to the extensive meanings both of planet and element. 
Every iron steam-ship which is made or destroyed adds ta 
or subtracts from the extensive meaning of the name 
steam-ship, without necessarily affecting the intensive 
meaning. Stage coach means as much as ever in one way, 
but in extension the class is nearly extinct. Chinese 
railway, on the other hand, is a term represented only by a 
single instance ; in twenty years it may be the name of a 
large class. 



The general principle of inference having been explained 

provided we have now before us the comparatively eas; 
task of tracing out the most common and impor Int fo'^s 
of deductive reasoning. The general problem of dedi^ 
tion IS a^ follows :-i^r.,;t one%r moreprM^^^^^ 
premuesto draw such otJur propositions iZu^es^^S ^^ 
be true wjun the premises are trvf. By deduction ZTyZi ) ^ 
gate and unfold the information contained 'nX^^^^^ 
and this we can do by one single vnle^For anyteZZ^r f ^ 
nru; in any proposition substitute the term whZ TaM ^ 
xnany premie to he identical unth it. To obtaTn cert^Q 


nf tI. IV- ! ^ ^'""^ '""^ "^® *^^ second and third Laws 

Jnatrect Deduction. In the present chapter however T 
shall confine my attention to those resX which cln^^ 

T^^^xCillK^^^ '^'"^^'^T "^^ ^"^^ of substitution 

system not on^v X "" • "^"^ ^°i^^^" ^^^^ «°^ harmonious 
system, not only the various moods of the ancient qvllnmor» 

biU a great number of equaUy important form of rj^^^^^^^ 
which had no recognised plL in the oldTo^c wT^^^^ 

apparatus 01 logical rules and mnemonic lines which 







!li ' 

i i'( 

Immediate Inference, 

Probably the simplest of all forms of inference is that 
which has been called Immediate Inference, because it can 
be performed upon a single proposition. It consists in 
joining an adjective, or other qualifying clause of the same 
nature, to both sides of an identity, and asserting the 
equivalence of the terms thus produced. For instance, 

Conductors of electricity = Non-electrics, 
it follows that 

Liquid conductors of electricity = Liquid non-electrics. 
If we suppose that 

Plants = Bodies decomposing carbonic acid, 
it follows that 

Microscopic plants = Microscopic bodies decotnposiug 
carbonic acid. 
In general terms, from the identity 

A = B 
we can infer tbe identity 

AC = BC. 
This is but a case of plain substitution; for by the rirst 
Law of Thought it must be admitted that 

AC = AC, 
and if, in the second side of this identity, we substitute 
for A its equivalent B, we obtain 

AC = BC. 
In like manner from the partial identity 

A = AB 
we may obtain 

by an exactly similar act of substitution ; and in every 
other case the rule will be found capable of verification by 
the principle of inference. The process when performed as 
here described will be quite free from the liability to error 
which I have shown ^ to exist in " Immediate Inference by 
added Determinants," as described by Dr. Thomson.^ 

^ El&nuntary Lesions in Logic, p. 86. 
' Outline of the L%tf« of Thought, § 87 

Inference mth Tioo Simple Identities. 

One of the most common forms of inference, and one to 
which I shall especially direct attention, is practised with 
two simple identities. From the two statements that 
" London is the capital of England " and " London is the 
most populous city in the world," we instantaneously draw 
the conclusion that " The capital of England is the most 
populous city in the world." Similarly, from the identities 
Hydrogen = Substance of least density, 
Hydrogen = Substance of least atomic weight, 
we infer '^ 

Substance of least density = Substance of least atomic 

The general form of the aigument is exhibited in the 

B = A (,) 

B = 2 

hence A=C. (3) 

We may describe the result by saying that terms identi- 
cal with the same term are identical with each other; and 
It IS impossible to overlook the analogy to the first axiom 
of Euclid that " things equal to the same thing are equal 
to each other." It has been very commonly supposed that 
this IS a fundamental principle of thought, incapable of 
reduction to anything simpler. But I entertain no doubt 
that this form of reasoning is only one case of the general 
rule of inference. We have two propositions, A = B and 
B = G, and we may for a moment consider the second one 
as affirming a truth concerning B, wliile the former one 
informs us that B is identical with A ; hence by substitu- 
tion we may affirm the same truth of A. It happens in 
this particular case that the truth affirmed is identity to 
G, and we might, if we preferred it, have considered the 
substitution as made by means of the second identity in 
tiie first. Having two identities we have a choice of the 
mode in which we will make the substitution, though thf 
result IS exactly the same in either case. 
Now compare the three following formulse 
(i) A = B = C, hence A = C 

(2) A = B - C, hence A - G 

(3) A '*' B -^ C, no infei-ence. 

« \ 




1 i I 

In the second formula we have an identity and a differ- 
ence, and we are able to infer a difference ; in the third we 
have two differences and are unable to make any inference 
at all. Because A and C both differ from B, we cannot 
tell whether they will or will not differ from each other, 
rhe flowers and leaves of a plant may both differ in colour 
from the earth in which the plant grows, and yet they may 
differ from each other ; in other cases the leaves and stem 
may both differ from the soil and yet agree with each other. 
Where we have difference only we can make no inference ; 
Avhere we have identity we can infer. This fact gives great 
countenance to my assertion that inference proceeds always 
through identity, but may be equally well effected in pro- 
positions asserting difference or identity. 

Deferring a more complete discussion of this point, I 
will only mention now that arguments from double identity 
occur very frequently, and are usually taken for granted, 
owing to their extreme simplicity. In regard to the equi- 
valence of words this form of inference must be constantly 
employed. If the ancient Greek 'voKko^ is our copper ^ then 
it must be the French cuivre, the German kup/er, the Latin 
cuprum^ because these are words, in one sense at least, 
equivalent to copper. Whenever we can give two defini- 
tions or expressions for the same term, the formula applies ; 
thus Senior defined wealth as " All those things, and those 
things only, which are transferable, are limited in supply, 
and are directly or indirectly productive of pleasure oi 
preventive of pain." Wealth is also equivalent to " things 
which have value in exchange ; " hence obviously, " things 
which have value in exchange = all those things, and those 
things only, which are transferable, «&;c." Two expressions 
for the same term are often given in the same sentence, and 
their equivalence implied. Thus Thomson and Tait say,^ 
^The naturalist may be content to know matter as that 
which can be perceived by the senses, or as that which 
3an be acted upon by or can exert force." I take this to 
mean — 

Matter = what can be perceived by the senses ; 
Matter =« what can be acted upon by or can exert 

I Trealvm on Naturtil PkUouophyt voL i. |>. l6l. 




For the term ''matter" in either of these identities we 
may substitute its equivalent given in the other definition. 
Elsewhere they often employ sentences of the form exem- 
plified in the following:* "The integral curvature, or 
whole change of direction of an arc of a plane curve, is the 
angle through which the tangent has turned as we pass from 
one extremity to the other." This sentence is certainly of 
the form — 

The integral curvature = the whole change of direc- 
tion, &c. = the angle through which the tangent 
has turned, &c. 
Dis^'uised cases of the same kind of inference occur 
throughout all sciences, and a remarkable instance is found 
in algebraic geometry. Mathematicians readily show that 
every equation of the form y = mx -{■ c corresponds to or 
represents a straight line ; it is also easily proved that the 
same equation is equivalent to one of the general form 
Ac -I- By 4- C = o, and vice versd. Hence it follows that 
every equation of the form in question, that is to say, 
every equation of the first degree, corresponds to or 
represents a straight line.* 

In/errnce with a Simple and a Partial Identity. 

A form of reasoning somewhat different from that last 
considered consists in inference between a simple and a 
partial identity. If we have two propositions of the forms 

A = B, 

B = BC, 
we may then substitute for B in either proposition its 
equivalent in the other, getthig in both cases A = BC ; 
in this we may if we like make a second substitution for 
l>i getting 

A = AC. 

Thus, since « The Mont Blanc is the highest mountain in 
il-urope, and the Mont Blanc is deeply covered with snow ' 
we infer by an obvious substitution that "The highest 
mountain in Europe is deeply covered with snow." These 
propositions when rigorously stated faU into the forms 
above exhibited. 

This mode of inference is constantly employed when foi 

\ Tv!?u*"** ^ Natural Philosophy, vol. i. p. 6. 
lodiiuuter's Flaw dj-ordimU G&jwary, chap. ii. pp. ii— 14 • 





If l'*i 




a term we substitute its definition, or vice vend. The very 
purpose of a definition is to allow a single noun to be 
employed in place of a long descriptive phrase. Thus, 
when we say " A circle is a curve of the second degree," we 
may substitute a definition of the circle, getting " A curve, 
all points of which are at equal distances from one point, is 
a curve of the second degree." The real forms of the pro- 
positions here given are exactly those shown in the sym- 
bolic Statement, but in this and many other cases it will be 
sufficient to state them in ordinary elliptical language for 
sake of brevity. In scientific treatises a term and its 
definition are often both given in the same sentence, as in 
" The weight of a body in any given locality, or tho force 
with which the earth attracts it, is proportional to its 
mass." The conjunction or in this statement gives the 
force of equivalence to the parenthetic phrjise, so that tlie 
propositions really are 

Weight of a body = force with which the eartli 
attracts it 

Weight of a body = weight, &c. proportional to its 
A slightly different case of inference consists in substitut- 
ing in a proposition of the form A = AB, a definition of the 
term B. Thus from A = AB and B =- C we get A = AC. 
For instance, we may say that " Metals are elements " and 
" Elements are incapable of decomposition." 

Metal = metal element. 

Element = what is incapable of decomposition. 

Metal = metal incapable of decomposition. 
It is almost needless to point out that the form of these 
arguments does not sufTer any real modification if some 
Df the terms happen to be negative ; indeed in the last 
example " incapable of decomposition " may be treated as 
a negative term. Taking 

A = metal C = capable of decomposition 

B = element c = incapable of decomposition ; 
ihe propositions are of the forms 

A = AB 
B = c 
wnence, by substitution. 

A -- A£. 




Infereiice of a Partial from Two Partial Identities. 

However common be the cases * of inference already 
noticed, there is a form occurring almost more frequentlv, 
and which deserves much attention, because it occupied'a 
prominent place in the ancient syllogistic system That 
system strangely overlooked all the kinds of argument we 
have as yet considered, and selected, as the type of all 
reasoning, one which employs two partial identities as 
premises. Thus from the propositions 

Sodium is a metil (i) 

Metals conduct electricity, (2) 

we may conclude that 

Sodium conducts electricity. (3) 

Taking A, B, C to represent the three terms respectively, 

the premises are of the forms 

A=AB (I) 

B ^ BC. (2) 

Now for B m (i) we can substitute its expression as given 

in (2), obtaining 

A = ABC, (3) 

or, in words, from 

Sodium =r sodium metil, (i\ 

Metal = metil conducting electricity, (2) 

we infer 

Sodium = sodium metal conducting electricity, (3 
which, m the elliptical language of common life, becomes 
" Sodium conducts electricity." 
The above is a syllogism in the mood called Barbara ^ in 
the truly barbarous language of ancient logicians ; and the 
nret hgure of the syllogism contained Barbara and three 
other moods which were esteemed distinct forms of argu- 
ment But it is worthy of notice that, without any real 
Change in our form of inference, we readily include these 
three other moods under Barbara. The negative mood 
telarent wiU be represented by the example 

Neptune is a planet, (i) 

No planet has retrograde motion ; (2) 

iience Neptune has not retrograde motion. (3) 

Willi!! T^^T-^'"''' ""^ ^^'^ ^"'^ other technical terras of the old loric 




If we put A for Neptune, B for planet, and C for " having 
retrograde motion," then by the corresponding negative 
term c, we denote "not having retrograde motion." The 
premises now fall into the forms 

A = AB (,) 

B = Be, (2) 

and by substitution for B, exactly as before, we obtain 

A = ABc (3) 

What is called in the old logic a particular conclusion 
may be deduced without any rcal variation in the symbols. 
Particular quantity is iodicated as before mentioned 
(p. 41), by joining to the term au indefinite adjective of 
quantity, such as sojtie, a part of, certain, &c., meaning that 
an unknown part of the term enters into the proposition 
as subject. Considerable doubt and ambiguity arise out of 
the question whether the part may not in some cases be 
the whole, and in the syllogism at least it must be under- 
stood in this sense.^ Now, if we take a letter to represent 
this indefinite part, we need make no change in our 
fornmlae to express the syllogisms Darii and Feria Con- 
uder the example — 

Some metals are of less density than water, (i) 
All bodies of less density than water will float 

upon the surface of water ; hence (2) 

Some metals will float upon the suiface of 
Let A = some metals, 

B = body of less density than water, 
C = floating on the surface of water 
then the propositions are evidently as before 

A = AB, 
B = BC; 
hence A = ABC, 

Thus the syllogism Darii does not really differ from^l(ar- 
bara. If the reader prefer it, we can readily employ a 
distinct symbol for the indefinite sign of quantity. 
Let P = some, 

Q = metal, 
B and C having the same meanings as before. Then the 
premises become 



* SUmmtary Leuom in LogUf pp. 67, 79. 




PQ = PQB, (,) 

B = BC; (2) 

hence, by substitution, as before, 

PQ = PQBC. (3) 

Except that the formulas look a little more complicated 
there is no difference whatever. 

The mood Ferio is of exactly the same character as 
Darii or Barbara, except that it involves the use of a 
negative term. Take the example, 

Bodies which are equally elastic in all directions do 

not doubly refract light ; 
Some crystals are bodies equally elastic in all direc- 
tions; therefore, some crystals do not doubly 
refract light. 
Assigning the letters as follows : — 
A = some crystals, 

B = bodies equally elastic in all directions, 
C = doubly refracting light, 
c = not doubly refracting liglit. 
Our argument is of the same form as before, and may 
be concisely stated in one line, 

A = AB = ALc. 
If It IS preferred to put PQ for the indefinite same crystals 
we have 

PQ - PQB = PQBc. 
Ihe only diflerence is that the negative terra c takes the 
place of C in the mood Darii 

Ellipsis of Terms in. Partial Identities. 

The reader will probably have noticed that the conclu- 
sion which we obtain from premises is often more full than 
that drawn by the old Aristotelian processes. Thus from 
bodium IS a metal," and « Metals conduct electricity," we 
inferred (p. 55) that - Sodium = sodium, metal, conduct^ 
lu^. „ o^^^'^^*^>'>" whereas the old logic simply concludes 
that Sodium conducts electricity." SymboHcally, from 
A = AB, and B = BC, we get A = ABC, whereas the old 
logic gets at the most A = AC. It is therefore well to 
snow tliat without employing any other principles of 

A - 7nV^^'^" *^^'^^^ ''^^^'^^^y described, we may infer 
A - AO trom A = ABC, though we cannot infer the latter 

i \ 






more full and accurate result from the former. We may 
show this most simply as follows : — 

By the first Law of Thought it is evident that 

AA = AA; 
and if we have given the proposition A = ABC, we may 
substitute for both the A's in the second side of the above, 

AA = ABC . ABC. 
But from the property of logical symbols expressed in the 
Law of Simplicity (p. 33) some of the repeated lettera may 
be made to coalesce, and we have 

A = ABC . C. 

Substituting again for ABC its equivalent A, we obtain 

A = AC, 
tlie desired result. 

By a similar process of reasoning it may be shown that 
we can always drop out any term appearing in one member 
of a proposition, provided that we substitute for it the 
v/hole of the other member. This process was described in 
my first logical Essay,^ as Intrinsic Mimination, but it 
might perhaps be better entitled the Ellipsis of Terms. 
It enables us to get rid of needless terms by strict 
substitutive reasoning. 

Inference of a Simple from Two Partial Identities. 

Two terms may be connected together by two partial 
identities in yet another manner, and a case of inference 
then arises which k of the highest importance. In the 
two premises 

A = AB (i) 

B = AB (2) 

the second member of each is the same ; so that we can by 
obvious substitution obtain 

A = B. 
Thus, in plain geometry we readily prove that " Every 
equilateral triangle is also an equiangular triangle," and we 
can with equal ease prove that " Every equiangular triangle 
is an equilateral triangle.' Thence by substitution, as 
explained above, we pass to the simple identity. 

Equilateral triangle = equiangular triangle. 

' Fure Logic, p. 19. 




We thus prove that one class of triangles is entirely 
identical with another class; that is to say, they differ 
only m our way of naming and regarding them. 

The great importance of this process of inference arises 
from the fact that the conclusion is more simple and general 
than either of the premises, and contains as much informa- 
tion as both of them put together. It is on this account 
constantly employed in inductive investigation, as wiU 
afterwards be more fully explained, and it is the natural 
mode by which we arrive at a conviction of the truth of 
simple identities as existing between classes of numerous 

Inference of a Limited from Two Partial Identities. 

We have considered some arguments which are of the 
type treated by Aristotle in the first figure of the syllogism 
But there exist two other types of argument which employ 
a pair of partial identities. If our premises are as shown 
in these symbols, 

B = AB (X) 

B = CB, U 

we may substitute for B either by (i) in (2) or by (2) in 
(I), and by both modes we obtain the conclusion 

AB = CB, (.) 

a proposition of the kind which we have caUed a limited 
Identity (p. 42). Thus, for example, 

Potassium = potassium metal (i) 

Potassium = potassium capable of floating on 

hence ' ^^^ 

Potassium metal = potassium capable of float- 
Tk- • ^?,g^n water. /x 

Ihis IS really a syllogism of the mood Darapti in the tliird 
fagure, except that we obtain a conclusion ot' a more exact 
chamcter than the old syllogism gives. From the premises 
Po assium is a metal ^ and "Potassium floats on water," 
Aristotle would have inferred that "Some metals float on 

metik". .1 '''^''''^ "^^"^ "^^^^ ^h*<^ the "some 
T^lL .^'•*^>„t^^e^a«swer would certainly be "Metal which 
^potassium " Hence Aristotle's conclusion simply leaves 
out some of the information afforded in the premises • it 







even leaves us open to interpret the scmit metals in a wider 
sense than we are warranted in doing. From these distinct 
defects of the old syllogism the process of substitution is 
free, and the new process only incurs the possible objection 
of being tediously minute and accurate. 

Miscellaneous Forms of Deductive Inference. 

The more common forms of deductive reasoning having 
been exhibited and demonstrated on the principle of 
substitution, there still remain many, in fact an indefinite 
number, whicli may be explained with nearly equal ease. 
Such as involve the use of disjunctive propositions will be 
described in a later chapter, and several of the syllogistic 
moods which include negative terms will be more con- 
veniently treated after we have introduced the symbolic 
use of the second and third laws of thought. 

We sometimes meet with a chain of propositions which 
allow of repeated substitution, and form an argument 
called in the old logic a Sorites. Take, for instance, the 

Iron is a metal, (i) 

Metals are good conductors of electricity, (2) 

Good conductors of electricity are useful for 

telegraphic purposes. (3) 

It obviously follows that 

Iron is useful for telegraphic purposes. (4) 

Now if we take our letters thus, 

A = Iron, B = metal, C = good conductor of 
electricity, D = useful for telegraphic purposes, 
the premises will assume the forms 

A = AB, (I) 

B = BC, (2) 

C = CD. (3) 

For B in (i) we can substitute its equivalent in (2) 
obtaining, as before, 

A = ABC. 
Substituting for C in this intermediate result its equivalent 
as given in (3), we obtain the complete conclusion 

A = ABCD. (4) 

The full interpretation is that Iron is iron, m,etal, good 
conductor of electricity ^ usefvX for telegraphic purposes, which 




is abridged in common language by the ellipsis of the 
circumstances which are not of immediate importance. 

Instead of all the propositions being exactly of the same 
kind as in the last example, we may have a series of 
premises of various character ; for instance. 

Common salt is sodium chloride, (i) 

Sodium chloride crystallizes in a cubical form, (2) 
What crystallizes in a cubical form does not 
possess the power of double refraction : (7 
it will foUow that • ^^ 

Common salt does not possess the power of double 
refraction. u) 

Taking our letter-terms thus, 
A = Common salt, 
B = Sodium chloride, 
C = Crystallizing in a cubical form, 
D = Possessing the power of double refraction, 
. we may state the premises in the forms 

A = B, (,) 

B = BC, I2) 

C = Cd, (3) 

Substituting by (3) in (2) and then by (2) as thus altered 
m (i) we obtain 

A = BCrf, (4) 

which is a more precise version of the common conclusion. 
We often meet with a series of propositions describing 
the qualities or circumstances of the one same thing, and 
we may combine them all into one proposition by the 
process of substitution. This case is, in fact, that which 
Or. Thomson has called "Immediate Inference by the 
sum of several predicates," and his example will serve my 
purpose well} He describes copper as "A metal— of a 
red colour— and disagreeable smell— and taste— all the 
preparations of which are poisonous— which is highly 
"lalleable— ductile— and tenacious— with a specific gravity 
of about 8.83." If we assign the letter A to copper, and the 
succeedmg letters of the alphabet in succession to the series 
of predicates, we have nine distinct statements, of the form 

A = AB (I) A = AC (2) A = AD (3) A = AK (9). 

we can readily combine these propositions into one by 

' Ah OuiUnt of the Necessary Lam of Thought, Filth Ed. p. 161. 






substituting for A in the second side of (i) its expression 
in (2). We thus get 

A = ABC, 
and by repeating the process over and over again we 
obviously get the single proposition 

A = ABCD . . JK. 
But Dr. Thomson is mistaken in supposing that we can 
obtain in this manner a definition of copper. Strictly 
speaking, the above proposition is only a description of 
copper, and all th« ordinary descriptions of substances in 
scientific works may be summed up in this form. Thus we 
may assert of the organic substances called Paraffins that 
they aie all saturated hydrocarbons, incapable of unitijig 
with other substances, produced by heating the alcoholic 
iodides with zinc, and so on. It may be shown that no 
amount of ordinary description can be equivalent to a de- 
finition of any substance. 


I have hitherto been engaged in showing that all the 
forms of reasoning of the old syllogistic logic, and an 
indefinite number of other forms in addition, may be 
readily and clearly explained on the single principle of 
^^ substitution. It is now desirable to show that the same 
U principle will prevent us falling into fallacies. So long 
as we exactly observe the one rule of substitution of 
equivalents it will be impossible to commit a paralogism, 
that is to break any one of the elaboi-ate rules of the 
ancient system. The one new rule is thus proved to be as 
powerful as the six, eight, or more rules by wliich the cor- 
rectness of syllogistic reasoning was guarded. 

It was a fundamental rule, for instance, that two nega- 
tive premises could give no conclusion. If we take the 

Granite is not a sedimentary rock, (l) 

Basalt is not a sedimentary rock, (2) 

we ought not to be able to draw any inference concerning 
the relation between granite and basalt. Taking our 
letter-terms thus : 

A = granite, B = sedimentary rock, C -= basalt, 
the premises may be expressed in the forms 



A - B, (1) 

C - B. (2) 

We have m this form two statements of difference; but 
the principle of inference can only work with a statement 
of agreement or identity (p. 63). Thus our rule gives 
us no power whatever of drawing any inference ; this is 
exactly in accordance with the fifth rule of the syllogism. 
It is to be remembered, indeed, that we clainf the 
power of always turning a negative proposition into an 
affirmative one (p. 45) ; and it might seem that the old rule 
agamst negative premises would thus be circumvented. 
Let us try. The premises (i) and (2) when affirmatively 
stated take the forms 

A = Aft (I) 

C = Cb. (2) 

The reader will find it impossible by the rule of substitu- 
tion to discover a relation between A and C. Three terms 
occur m the above premises, namely A, b, and C ; but they 
are so combmed that no term occurring in one has its 
exact equivalent stated in the other. No substitution 
can therefore be made, and the principle of the fifth rule of 
the syllogism holds true. Fallacy is impossible. 

It would be a mistake, however, to suppose that the 
mere occurrence of negative terms in both premises of a 
syllogism renders them incapable of yielding a conclusion. 
Ihe old rule informed us that from two negative premises 
DO conclusion could be drawn, but it is a fact that the rule 
m this bare form does not hold universaUy true • and I 
am not awai-e that any precise explanation has been Lnven 
of the conditions under which it is or is not imperative, 
l^onsider the following example : 

Whatever is not metallic is not capable of power- 
ful magnetic influence, d) 
Carbon is not metallic, )2) 
Therefore, carbon is not capable of powerful man- 
netic influence. °/^x 
r^rtr ^ff/^^ distinctly negative premises (i) and 
sfon r^f ^l 7^ ^'^^^ * perfectly valid negative conclu- 
sion (3). The syllogistic rule is actually falsified in its bare 
and general statement In this and many other cases we 
can convert the propositions into affirmative ones which will 
yield a conclusion by substitution without any difficulty 




u li) 


^ m 









To show this let 

A = carbon, B = metallic, 

C = capable of powerful magnetic influence. 
The premises readily take the fonns 

6 = 6c, (f ; 

A = A6, (2) 

and substitution for h in (2) by means of (i) gives the 

conclusion . ^ 

A = Ahc. (3) 

Our principle of inference then includes the nile of 
negative premises whenever it is true, and discriminates 
correctly between the cases where it does and does not 

hold true. ^ rr j- 

The paralogism, anciently called the Fallacy of Undts- 
trihuted Middle, is also easily exhibited and infallibly 
avoided by our system. Let the premises bo 

Hydrogen is an element, (l J 

All metals are elements. (2) 

According to the syllogistic rules the middle term "element " 
is here undistributed, and no conclusion can be obtamed ; 
we cannot tell then whether hydrogen is or is not a metal 
Represent the terms as follows 

A = hydrogen, 
B = element, 
C = metal. 

The premises then become 

*^ A = AB, (i^ 

C = CB. (2) 

The reader will here, as in a former page (p. 62), find it 
impossible to make any substitution. The only term which 
occurs in both premises is B, but it is differently combined 
in the two premises. For B we must not substitute A, 
which is equivalent to AB, not to B. Nor must we confuse 
together CB and AB, which, though they contain one coin- 
mon letter, are different aggregate terms. The ride ot sub- 
stitution gives us no right to decompose combinations ; 
and if we adhere rigidly to the rule, that if two terms are 
stated to be equivalent we may substitute one for the other, 
we cannot commit the fallacy. It is apparent that the form 
of premises stated above is the same as that which wc 
obtained by translating two negative premises mto the 
affirmative form. 



The old fallacy, technically called the Illicit Process of 
the Major Term, is more easy to commit and more difficult 
to detect than any other breach of the syllogistic rules. In 
our system it could hardly occur. From the premises 
All planets are subject to gravity, (i) 

Fixed stars are not planets, (2) 

we might inadvertently but fallaciously infer that, " Fixed 
stars are not subject to gravity." To reduce the premises 
to symbolic form, let 

A = planet 
J^ = fixed star 
C = subject to gravity ; 
then we have the propositions 

A = AC (I) 

B = Ba. (2) 

The reader will try in vain to produce from these premises 
by legitimate substitution any relation between B and C ; 
he could not then commit the fallacy of asserting that B is 
not G. 

There remain two other kinds of paralogism, commonly 
known as the fallacy of Four Terms and the Illicit Process 
of the Minor Teim. They are so evidently impossible 
while we obey the rule of the substitution of equivalents, 
that it is not necessary to give any illustrations. When 
there are four distinct terms in two propositions a.s in 
A = B and C = U, tiiere cuuld evidently ije no opening for 
substitution. As to the Illicit Process of the Minor Terra 
it consists in a flagrant substitution for a term of another 
wider term which is not known to be. equivalent to it, 
and which is therefore not allowed by our rule to be 
•ubstituted for it 

CHAP, v.] 






In the previous chapter I have exhibited various cases 
of deductive reasoning by the process oi substitution, avoid- 
ing the introduction of disjunctive propositions ; but we 
cannot long defer the consideration of this more complex 
class of identities General terms arise, as we liave seen 
(p. 24), from classifying or mentally uniting together all 
objects which agree in certain qualities, the value of this 
union consisting in the fact that the power of knowledge 
is multiplied thereby. In forming such classes or general 
notions, we overiook or abstract the points of difference 
which exist between the objects joined together, and fix our 
attention only on the points of agreement But every 
process of thought may be said to have its inverse process, 
wliich consists in undoing the effects of the direct process. 
Just as division undoes multiplication, and evolution un- 
does involution, so we must have a process which undoes 
generalization, or the operation of forming general notions. 
This inverse process will consist in distinguishing the 
separate objects or minor classes which are the constituent 
parts of any wider class. If we mentally unite together 
certain objects visible in the sky and call tliem planets, we 
shall afterwards need to distinguish the contents of this 
general notion, which we do in the disjunctive proposi- 
tion — 

A planet is either Mercury or Venus or the Earth or 

or Neptune. 

Having formed the very wide class " vertebrate animal," 
we may specify its subordinate classes thus : — " A verte- 

brate animal is either a mammal, bird, reptile, or fish." 
Nor is there any limit to the number of possible altema 
tives. "An exogenous plant is either a ranunculus, a 
poppy, a cnicifer, a rose, or it belongs to some one of -the 
other seventy natui-al orders of exogens at present recog- 
nized by botanists." A cathedral church in England must 
be either that of London, Canterbury, Winchester, Salis- 
bury, Manchester, or of one of about twenty-four cities 
possessing such churches. And if we were to attempt to 
specify the meaning of the term " star," we should require 
to enumerate as alternatives, not only the many thousands 
of stars recorded in catalogues, but the many millions un- 

Whenever we thus distinguish the parts of a general 
notion we employ a disjunctive proposition, in at least one ^^ 
side of which are several alternatives joined by the so- 
called disjunctive conjunction or, a contracted form of other. 
There must be some relation between the parts thus con- 
nected in one proposition ; we may call it the disjwnctive or 
alternative relation, and we must carefully inquire into its 
nature. This relation is that of ignorance and doubt, 
giving rise to choice. Whenever we classify and abstract 
we must open the way to such uncertainty. By fixing our 
attention on certain attributes to the exclusion of others 
we necessarily leave it doubtful what those other attributes 
are. The term " molar tooth " bears upon the face of it 
that It is a part of the wider term " tooth." But if we 
meet with the simple term " tooth " there is nothing to in- 
dicate whether it is an incisor, a canine, or a molar tooth. 
Ihis doubt, however, may be resolved by further informa- 
tion, and we have to consider what are the appropriate 
logical processes for treating disjunctive propositions in 
connection with other propositions disjunctive or otherwise. 

Expression of tlie Alternative Relation. 

In order to represent disjunctive propositions with con- 
venience we require a sign of the alternative relation, 
equivalent to one meaning at least of the little conjunc- 
tion or so frequently used in common language. I pro- 
pose to use for this purpose the symbol .|. . In my first 
logical essay I followed the practice of Boole and adopted 

F 2 










the sign +; but this sign should not be employed unless there 
exists exact analogy between mathematical addition and 
logical alternation. We shall find that the analogy is im- 
perfect, and that there is such profound difference between 
logical and mathematical terms as should prevent our 
uniting them by the same symbol. Accordingly I have 
chosen a sign •!• , which seems aptly to suggest whatever 
degree of analogy may exist without implying more. 
The exact meaning of the symbol we will now proceed to 

Nature of the Alter native Relation. 

Before treating disjunctive propositions it is indispens- 
able to decide whether the alternatives must be considered 
exclusive or unexclusive. By exclusive aitemativcs we 
mean those which cannot contain the same things. If we 
say " Arches are circular or pointed," it is certainly to be 
understood that the same arch cannot be described as both 
circular and pointed. Many examples, on the other hand, 
3an readily be suggested in which two or more alteraatives 
may hold true of the same object. Thus 

Luminous bodies are self-luminous or luminous by 
It is undoubtedly possible, by the laws of optics, that the 
3ame surface may at one and the same moment give ofl* 
light of its own and reflect light from other bodies. We 
speak familiarly of cka/or dumb persons, knowing that the 
majority of those who are deaf from birth are also dumb. 

There can be no doubt that in a great many cases, 
perhaps the greater number of cases, alternatives are 
exclusive as a matter of fact. Any one number is 
incompatible with any other ; one point of time or place 
is exclusive of all others. Roger Bacon died either in 
1284 or 1292 ; it is certain ^hat he could not die in both 
years. Henry Fielding was born either in Dublin or 
Somersetshire; he could not be born in both places. 
There is so much more precision and clearness in the use 
ot exclusive alternatives that we ought doubtless to select 
them when possible. Old works on logic accordingly 
wntained a rule directing that the Membra divideniia, the 




parts of a division or the constituent species of a genus, 
should be exclusive of each other. 

It is no doubt owing to the great prevalence and con- 
venience of exclusive divisions that the majority of logi- 
cians have held it necessary to make every alternative in 
a disjunctive proposition exclusive of every other one 
Aquinas considered that when this was not the case the 
proposition was actually false, and Kant adopted the 
same opinion.* A multitude of statements to the same 
eHect might readily be quoted, and if £he question were 
to be determined by the weight of historical evidence 
It would certainly go against my view. Among recent 
logicians Hamilton, as well as Boole, took the exclusive 

V^^ i^".*^r *^®^ *^® authorities to the opposite effect. 
A\hately, Mausel, and J. S. Mill have all pointed out that 
we may often treat alternatives as Compossible, or true at 
the same time. Whately gives us an example,^ " Virtue 
tends to procure us either the esteem of mankind, or the 
favour of God," and he adds—" Here both members are 
true, and consequently from one being aftirmed we are not 
authorized to deny the other. Of course we are left to 
conjecture in each case, from the context, whether it is 
meant to be implied that the members are or are nor 
exclusive." Mansel says,^ " JVe mai/ happen to know that 
two alternatives cannot be true together, so that the 
athrmation of the second necessitates the denial of the 
hrst ; but this, as I^thius observes, is a material, not a 
formal consequence." Mill has also pointed out the 
absurdities which would arise from always interpreting 
alternatives as exclusive. « If we assert," he says,* " that 
a man who has acted in some particular way must be 
cither a knave or a fool, we by no means assert, or intend 
to assert, that he cannot be both." Again, "to make an 
entirely unselfish use of despotic power, a man must be 
either a saint or a philosopher. Does the dis- 
junctive premise necessarily imply, or must it be construed 
as supposing, that the same person cannot be both a 

I MansePs Aldrich, p. 103, and ProUgomma Logica, p. 221. 
, KUmenU of Logic, Book II. chap. iv. sect. 4. 
^ Aldricb, Artis Logica: Budimenta, p. 104. 
t^ramiiiaium of Sir W. Hamilton's Philosophy, pp. 452-454. 


i J 


- t 

\ \ 





,NM f 






saint and a philosopher ? Such a construction would be 

I discuss this subject fully because it is really the point 

which separates my logical system from that of Boole. 

In his Laws of Thoiujht (p. 32) he expressly says, 

** In strictness, tlie words * and,' * or,' interposed Ijetween 

the terms descriptive of two or more classes of objects, 

imply that those classes are quite distinct, so that no 

member of one is found in another." This I altogether 

dispute. In the ordinary use of these conjunctions we do 

not join distinct terms only ; and when terms so joined 

do prove to be logically distinct, it is by virtue of a tacit 

•premise, something in the meaning of the names and 

our knowledge of them, which teaches us that they are 

distinct. If our knowledge of the meanings of the 

words joined is defective it will often be impossible 

to decide whether tenns joined by conjunctions are 

exclusive or not. 

In the sentence " Repentance is not a single act, but 
a habit or virtue," it cannot be implied that a virtue is 
not a habit ; by Aristotle's definition it is. Milton has the 
expression in one of his sonnets, " Unstain'd by gold or 
fee," where it is obvious that if the fee is not always gold, 
the gold is meant to be a fee or bribe. Tennyson has the 
expression " wreath or anadem." Most readers would be 
quite uncertain whether a wreath may be an anadem, or 
an anadem a wreath, or whether they are quite distinct or 
quite the same. From Darwin's Origin of Species, I 
take the expression, "When we see any part or organ 
developed in a remarkable degree or manner." In this, or 
is us«d twice, and neither time exclusively. For if part 
and organ are not synonymous, at any rate an organ is a 
part. And it is obvious that a part may be ileveloped at 
the same time both in an extraordinary degree and an 
extraordinary manner, although such cases may be com- 
paratively rare. 

From a careful examination of ordinary writings, it will 
tlms be found that the meanings of terms joined by "and," 
" or " vary from absolute identity up to absolute contrariety. 
There is no logical condition of distinctness at all, and 
when we do choose exclusive alternatives, it is because 
our subject demands it The matter, not the form of an 




expression, points out whether terms are exclusive or not.' 
In bills, policies, and other kinds of legal documents, it 
is sometimes necessary to express very distinctly that 

alternatives are not exclusive. The form — is then 

used, and, as Mr. J. J. Murphy has remarked, this form 
coincides exactly in meaning with the symbol .|. . 

In the first edition of this work (vol. i., p. 81), I took 
the disjunctive proposition " Matter is solid, or liquid, or 
gaseous," and treated it as an instance of exclusive altern- 
atives, remarking that the same portion of matter cannot be 
at once solid and liquid, property speaking, and that still less 
can we suppose it to be solid and gaseous, or solid, liquid, 
and gaseous all at the same time. But the experiments of 
Professor Andrews show that, under certain conditions of 
temperature and pressure, there is no abrupt change from 
the liquid to the gaseous state. The same substance may be 
m such a state as to be indiflerently described as liquid and 
gaseous. In many cases, too, the transition from solid to 
liquid is gradual, so that the properties of solidity are at least 
partially joined with those of liquidity. The proposition 
then, instead of being an instance of exclusive alternatives, 
«eems to afford an excellent instance to the opposite effect. 
When such doubts can arise, it is evidently impossible to 
treat alternatives as absolutely exclusive by the logical 
nature of the relation. It becomes purely a question of 
the matter of the proposition. 

The question, as we shall afterwards see more fully, is 
one of the greatest theoretical importance, because it 
concerns the true distinction between the sciences of 
Logic and Mathematics. It is the foundation of number 
that every unit shall be distinct from every other unit ; 
but Boole imported the conditions of number into the 
science of Logic, and produced a system which, though 
wonderful in its results, was not a system of logic at all. 

Laws of tlie Diy'unctive Relation. 

In considering the combination or synthesis of terms 
(P- 30), we found that certain laws, those of Simplicity 

> Pwc Logic, pp 76, 77. 

I mV^ 

"V ■,' 


' < 









and Commutativeness, must be observed. In uniting 
terms by the disjunctive symbol we shall find that the 
same or closely similar laws hold true. The all ^natives 
of either member of a disjunctive proposition are certainly 
commutative. Just as we cannot properly distinguish 
between rich and rare gems and rare and rich gems, so we 
must consider as identical the expression rich or rare gems, 
and rare or rich gems. In our symbolic language we may 


A + B = 15 + A. 
The order of statement, in short, has no effect upon the 
meaning of an aggregate of alternatives, so that the 
Law of Commutativeness holds true of the disjunctive 


As we have admitted the possibility of joining as alter- 
natives t^rms which are not really different, the question 
arises. How shall we treat two or more alternatives when 
they are clearly shown to be the same? If we have it 
asserted that P is Q or R, and it is afterwanls proved that 
Q is but another name for R, the result is that P is either 
K or R How shall we interpret such a statement ? What 
would be the meaning, for instance, of " wreath or anadem " 
if, on referring to a dictionary, we found anadem described 
as a wreath ? I take it to be self-evident that the meaning 
would then l)ecome simply "wreath." Acconlingly we 
may affinn the general law 

A + A = A, 
Any number of identical alternatives may always be 
reduced to, and are logically equivalent to, any one of 
those alternatives. This is a law which distinguishes 
mathematical terms from logical terms, because it obviously 
does not apply to the former. I propose to call it the Law 
of Unity, because it must really be involved in any 
definition of a mathematical unit This law is closely 
analogous to the Law of Simplicity, AA = A ; and the 
nature of the connection is worthy of attention. 

Few or no logicians except De Morgan have adequately 
noticed the close relation between combined and disjunctive 
terms, namely, that every disjunctive term is the negative 
of a corresponding combined term, and vice versd. Consider 

the term 

Malleable dense metal 




How shall we describe the class of things which are not 
malleable-dense-metals ? Whatever is included under that 
terni must have all the qualities of malleability, denseness 
and metalhcity. Wherever any one or more of the qualities 
IS wanting, the combined term will not apply. Hence the 
negative of the whole term is 

Not-malleable or not-dense or not-metallic. 
In the above the conjunction or must clearly be inter- 
preted as unexclusive; for there may readily be objects 
which arc both not-malleable, and not-dense, and ])ei-liaps 
not-metaUic at the same time. If in fact we were required 
to use or m a strictly exclusive manner, it would be 
requisite to specify seven distinct alternatives in order to 
describe the negative of a combination of three terms. 
I he negatives of four or five terms would consist of fifteen 
or thirty-one alternatives. This consideration alone is 
sufhcient to prove that the meaning of or cannot be 
always exclusive in common language. 

Expressed symbolically, we may say that the negative 

is not-A or not-B or not-C ; 

that is, a I- b f c. 

Reciprocally the negative of 

P + Q I- R 

Every disjunctive term, then, is the negative of a 
combined term, and mce versd. 

Apply this result to the combined term AAA, and its 
negative is 

a •{• a -j- a. 
Since AAA is by the Law of Simplicity equivalent to A 
so a jr a J- a must be equivalent to a, and the Law of 
the^o'th ^^^^' ^^^ ^*^ ^'^"^ necessarily presupposes 

Symbolic expression of the Law of Ditality. 

We naay now employ our symbol of alternation to 
express in a clear and formal manner the third Funda- 
mental Law of Thought, which I have called the Law 
ot Duality (p. 6). Taking A to represent any class or 









object or quality, and B any otlier class, object or quality, 
we may always assert that A either agrees with B, or does 
not agree. Thus we may say 

A = AB .|. Ab. 

This is a formula which will henceforth be constantly 
employed, and it lies at tlie basis of reasoning. 

The reader may perliaps wish to know why A is inserted 
in both alternatives of the second member of the identity, 
and why the law is not stated in the form 

A = B .|. b. 
But if he will consider the contents of the last section 
(p. 73), he will see that tlie latter expression cannot be 
correct, otherwise no term could have a corresponding 
negative term. For the negative of B .|. 6 is 6B, or a self- 
contradictory term ; thus if A were identical with B j. b 
its negative a would be non-existent. To say the least, 
this i-esult would in most cases be an absurd one, and I 
see much reason to think that in a strictly logical point ol 
view it would always be absurd. In all probability we 
ought to assume as a fundamental logical axiom that every 
iemi has its negative in thought. We cannot think at all 
without separating what we think about from other things, 
and these things necessarily form the negative notion.' 
It follows that any proposition of the form A = B J- 6 is 
just as self-contradictory as one of the form A = hb. 

It is convenient to recapitulate in this place the thru* 
Laws of Thought in their symbolic form, thus 
Law of Identity A = A. 

Law ol' (Juuirnuiutiou Au -= o. 

Law of Duality A = AB •!• Ab. 

Various Foiins of the Disjunctive Proposition. 

Disjunctive propositions may occur in a great variety of 
forms, of which the old logicians took insufficient notice. 
There may be any number of alternatives, each of which 
may be a combination of any number of simple terms. A 
proposition, again, may be disjunctive in one or both 
members. The proposition 

* Pure LogiCy p. 65. See also the criticism of this point by De 
Morgan in the Athenaum^ No. 1892, 30th January, 1864 ; p. 155. 

Solids or liquids or gases are electrics or conductors 
of electricity 
is an example of the doubly disjunctive form. The mean- 
mg of such a proposition is that whatever falls under any 
one or more alternatives on one side must fall under one 
or more alternatives on the other side. From what has 
been said before, it is apparent that the proposition 

A.|.B=C.|.D "^ 

will correspond to 

each member of the latter being the negative of a member 

ot the former proposition. 
As an instance of a complex disjunctive proposition I 

naay give Senior's definition of wealth, which; briefly 

stated, amounts to the proposition « Wealth is what is 

transferable, limited in supply, and either productive of 

pleasure or preventive of pain." * 

Let A = wealth 
B = transferable 
C = limited in supply 
D = productive of pleasure 

»,. , ^ E = preventive of pain. 

ihe definition takes the form 

K . r . A = BC(D.|.E); 

but If we develop the alternatives by a method to be 
afterwards more fully considered, it becomes 

A = BCDE .|. BCDe .|. BCrfE. 

foutiS TdJ^M ^^ ^ '™ .^^^ ^'^"^P^^^ proposition is 

thul, '^^ '^'"" ^^'^ ^^"^'^ ""^ *^^ ^'P^^^^^ ^ succession, 
' A = he 
B = rich 

C ■» absolutely mad 
D « weakness itself 
E = subjected to bad advjce 

« ^It' ^^^'^^P- '^- ^'l^^' P^re Logic, p. 69. 
^un ih. Syllogum, No. ,11. p. ,2. Camb. Phil, l^'n^ Vol. , 

■ \i 

i( :■ 





F = subjected to most unfavourable circumstances, 
the proposition will take the form 

A = AB{C I- D (E I- F)}, 
and if we develop the alternatives, expressing some of 
the diflerent cases which may happen, we obtain 

The above gives the strict logical interpretation of the 
sentence, and the first alternative ABC is capable of de- 
velopment into eight cases, according as 1), E and F are or 
are not present. Although from our knowledge of the 
matter, we may infer that weakness of character cannot be 
asserted of a person absolutely mad, there is no explicit 
statement to this eflect. 

Inference by Disjunctive Propositions. 

Before we can make a free use of disjunctive proposi- 
tions in the processes of inference we must consider how 
disjunctive terms can be combined together or with 
simple terms. In the first place, to combine a simple term 
with a disjunctive one, we must combine it with every 
alternative of the disjunctive term. A vegetable, for 
instance, is either a herb, a shrub, or a tree. Hence an 
exogenous vegetable is either an exogenous herb, or an 
exogenous shrub, or an exogenous tree. Symbolically 
stated, this process of combination is as follows, 

A(B.|.C) = ABlAC. 

Secondly, to combine two disjunctive terms with each 
other, combine each alternative of one with each alterna- 
tive of the other. Since flowering plants are either 
exogens or endogeus, and are at the same time either 
herbs, shrubs or trees, it follows that there are altogether 
six alternatives — namely, exogenous herbs, exogenous 
shrubs, exogenous trees, endogenous herbs, endogenous 
shrubs, endogenous trees. This process of combination is 
shown in the general form 

(A .|. B) (C .|. D .|. E) = AC I- AD .|. AE .| BC I- BD + BE 

It is hardly necessary to point out that, however 
numerous the terms combined, or the alternatives in those 
terms, we may effect the combination, provided each alter- 
native is combined with each alternative of the other 
terms, as in the algebraic process of multiplication. 




Some processes of deduction may be at once exhibited. 
We may always, for mstance, unite the same qualifyinff 
term to each side of an identity even though one or both 
members of the identity be disjunctive. Thus let 

A = B .|. C. 
Now it is self-evident that 

AD = AD. 

AD = BD + CD. 

Since a gaseous element is either hydrogen or owcen 
or nitiogen, or chlorine, or fluorine." it follows il.at "a free 
Saseous element ,s either free hydrogen, or free oxygen 
or free nitrogen, or free chlorine, or frJe fluorine " 

This process of combination will lead to most useful in- 
^^^Tf^ ^* qualifying adjective combined with both 
sides of the proposition is a negative of one or more alter- 
natives. Since chlorine is a coloured gas, we may infer 
hat "a colourless gaseous element is either (colourless 
hydrogen oxygen, nitrogen, or fluorine." The altematrvi 
chlonne disappears because colourless chlorine does not 
exist Again, since "a tooth is eitl^r an incisor, canine 
b cuspid, or molar. • it follows that -''^ not-incisor l^rS 
either canme. bicuspid, or molar." The geneml rule is that 
from the denial of any of the altemativl the afflrmaUon 
of the remainder can be inferred. Now this result c earfy 

S7-Xr '"^^ "' ^"^"*"««" = '- 'f - >^- 

evilrt iSty " '""^^^^''^^'^^^ «- -'^'^ of the self- 

A6 = Ab, 
we obtain AJ = AB6 .|. A5C I- A JD • 

and as the first of the three alternatives is self-contra 

»,. AJ = AJC .|. A6D. 

the"'lZ,rf^'" «''^r '?"'"*^"* '^"'' «^Pl«'"« "'"t mood of 

tivfsv^l" Urn %'l?r' "^"^^""y ''^'''- *»t th" Disjunc- 
tive byUogism of the mood ponendo tollens. which affirms 

■ 'i 









one alternative, and thence infers the denial of the rest, 
cannot be held true in this system. If I say, indeed, that 

Water is either salt or fresh water, 
it seems evident that " water which is salt is not fresh." 
But this inference really proceeds from our knowledge that 
water cannot be at once salt and fresh. This inconsistency 
of the alternatives, as I have fully shown, will not always 
hold. Thus, if I say 

Gems are either rare stones or beautiful stones, (i) 
it will obviously not follow that 

A rare gem is not a beautiful stone, (2) 

nor that 

A beautiful gem is not a rare stone. (3) 

Our symbolic method gives only true conclusions ; for if 
we take 

A = gem 
B = rare stone 
C = beautiful stone, 
the proposition (i) is of the form 

A = B .|. C 
hence AB = B I- BC 

and .^^ = BC.|.C; 

but these inferences are not equivalent to the false ones 
(2) and (3). 

We can readily represent disjunctive reasoning by the 
modus fonendo tollens, when it is valid, by expressing the 
inconsistency of the alternatives explicitly. Thus if we 
resort to our instance of 

Water is either salt or fresh, 
and take 

A = Water B = salt C = fresh, 
then the premise is apparently of the form 

A = ABl-AC; 
but in reality there is an unexpressed condition that " what 
is salt is not fresh," from which follows, by a process of 
inference to be afterwards described, that " what is fresh 
is not salt." We have then, in letter-terms, the two pro- 

B = B(j 
C = JC. 
If we substitute these descriptions in the original pre 
position, we obtain 

A = ABc .|. AhG ; 
uniting B to each side we infer 

AB = ABc .|. ABbG 
or AB = ABc ; 

that is, 

Water which is salt is water salt and not fresh. 
I should weary the reader if I attempted to illustrate 
the multitude of forms which disjunctive reasoning may 
take; and as in the next chapter we shall be constantly 
treating the subject, I must here restrict myself to a single 
instance. A very common process of reasoning consists in 
the determination of the name of a thing by the successive 
exclusion of alternatives, a process called by the old name 
abscissto mfimti. Take the case : 

Red-coloured metal is either copper or gold (i) 

Copper is dissolved by nitric acid (2) . 

This specimen is red-coloured metal (3) 

This specimen is not dissolved bv nitric acid (4) 

Therefore, this specimen consists" of gold f c) 
Let us assign the letter-symboU thus— 
A = this specimen D = gold 

B = red-coloured metal E = dissolved by nitric acid 
C = copper 

Assuming that the alternatives copper or (jold are 
intended to be exclusive, as just explained in the case of 

S ^^''^'' *^® P"^""^'^' ^^3^ ^ «^ted in the 

B = BCrf.|.BcD (i\ 

G = CE U 

1 : 1? (3) 

Substituting for C in (i) by means of (2) we get ^^^ 
^ ^ B = BCrfE -h Bel) 

i^rom (3) and (4) we may infer likewise 

A = AR; 
^Jfit'foKs'har'^""^ '" ^ '"^ equivalent. just 
^ A = ABGdEe I- ABcBe 

Uie first of the alternatives being contradictory the result 

A = AlicDe 

' Hi 





which contains a full description of " this specimen " as 
furnished in the premises, but by ellipsis asserts that it is 
gold. It will be observed that in the symbolic expression 
(l) I have explicitly stated what is certainly implied, that 
copper is not gold, and gold not copper, without which 
condition the inference would not hold good. 



The forms of deductive reasoning as yet considered, are 
mostly cases of Direct Deduction as distinguished from 
those which we are now about to treat. The method of 
Indirect Deduction may be described as that which points 
out what a thing is. by showing that it cannot be anything 
el3e. We can define a certain space upon a map, either by 
colouring that space, or by colouring all except the space ; 
the first, mode is positive, the second negative. The 
difference, it will be readily seen, is exactly analogous to 
that between the direct and indirect modes of proof in 
geometry. Euclid often shows that two lines are equal by 
showing that they cannot be unequal, and the proof rests 
jipon the known number of alternatives, greater, equal or 
loss, which (ire alone conceivable. In other cases, as for 
nisUmce m the seventh proposition of the first book, he 
Shows that two lines must meet in a particular point, by 
showing that they cannot meet elsewhere. 

In logic we can always define with certainty the utmost 
number of «altematives which are conceivable. The Law 
quality (pp. 6, 74) enables us always to assert that any 
?hc ?" «^,circumstance whatsoever is either present or 
X'. Whatever may be the meaning of the terms A 
•urn 15 It IS certainly true that 

A = AB.|.A5 
B = AB.|.aB. 

Teh Wo • M '""^^'^'^ ^^ ^^'^ P^^^l^"^> ^"^ ^J^ich e^ve 
«"tii invariable and necessary conditions of all thought, 


■ I 

\ V' 




that they need not be specially laid down. The Law of 
Contradiction is a further condition of all thought and of 
all logical symbols; it enables, and in fact obliges, us to 
reject from further consideration all terms which imply the 
presence and absence of the same quality. Now, when- 
ever we bring both these Laws of Thought into explicit 
action by the method of substitution, we employ the 
Indirect Method of Inference. It will be found that we 
can treat not ouly those arguments already exhibited 
according to the direct method, but we can include an 
infinite multitude of other arguments which are incapable 
of solution by any other means. 

Some philosophers, especially those of France, have held 
that the Indirect Method of Proof has a certain inferiority 
to the direct method, which should prevent our using it 
except when obliged. But there are many truths which 
we can prove only indirectly. We can prove that a 
number is a prime only by the purely indirect method of 
showing that it is not any of the numbers which^ have 
divisors, and the remarkable process known as Eratos- 
thenes* Sieve is the only mode by which we can select the 
prime numbers.^ It bears a strong analogy to the indirect 
method here to be described. We can prove that the side 
and diameter of a square are incommensurable, but only in 
the negative or indirect manner, by showing tliat the con- 
trary supposition inevitably leads to contradiction.* Many 
other demonstrations in various branches of the mathe- 
matical sciences proceed upon a like method. Now, if 
there is only one important truth which must be, and can 
only be, proved indirectly, we may say that the process is a 
necessary and sufficient one, and the question of its com- 
parative excellence or usefulness is not worth discussion. 
As a matter of fact I believe that nearly half our logical 
conclusions rest upon its employment. 

' SeeHorsley, Philosophical Transactioni, 1772 ; vol. Ixii. p. 327. 
Montucla, Histoire dts Matheviatuiwa, vol. i. p. 2^9. renny 
Cydopadicty article '* Eratosthenes." 

« Euclid, Book X. Prop. 117. 



Simple IllustrcUions. 
J\Sn%t^ 'l^ P""""? ''"'J '^"l'« of this method we 

has had th^lewt £cal ^i "•'*'"?«• ^"^ P^"^" ^»'<' 

dmw from tie abo^;^*v"^' " *^*'* t'"'* ^« <^^ 
one. namely, P^'P'^'tion an apparently different 

o^sldeZMf Sro^thr'^r ^^ ^"•«-'' have 
purely self-evident and neithTr '^^ P~P««'«on« ^ be 
analysis, a creat manv L^ ne^dmg nor aUowing 

while tekchfnrio^cLa^fiP!r"''u \^''^^ '^^'^^^ 

close connecti'on ^^^ ttrn'^lSelfaTr *^^ 
complete system of InrnV ^iii V ."Y^^^^e tJiat a true and 

this VeL. which Z^l cX r '^'■ ""^^^^ °f 
rem^; the full procesTisl^folwJ''^'"''^^ ^'«- 
lirstly. by the Uw of Duality we know that 

If .> i^°K'f""^^^^ Metal or Not-metal 

LL^ntid^tr,: s^ '' '' y *^«^-'"- - 

is an ekment and a n^L'^leS'X^ ''* ^"^ ^'''"g 
to the Law of ContradS ^ L-' "* «PPo«'tion 
other alteraative then 1 1 .V /According to the only 
metal. ' ^''' *''® "ot-elenient must be a not 

To represent this process of inference avmV^i;. ii 
take tlie premise in the form *""'® symbolicaUy we 

tTrdtria""* "' "" ^'^ ''' ^-"-'y *e term noS is 
^'^tS):SS^^'^^ ^^ -^escriptio^a, 



r ^i 








Hence it results that h is either nothing at all, or it is db; 
and the conclusion is 

As it will often be necessary to refer to a conclusion of 
this kind I shall call it, as is usual, the Cmtraposthve 
Proposition of the original. The reader need hardly be 
cautioned to observe that from all A's are B's it does not 
follow that all not-A's are not-B's. For by the Law of 
Duality we have 

and it will not be found possible to make any substitution 
in this by our original prendse A = AB. It still remains 
doubtful, therefore, whether not-metal is element or not- 

element. . . . , 

The proof of the Contrapositive Proposition given above 
is exactly the same as that which Euclid applies in the 
case of geometrical notions. De Morgan describes Euclid s 
process as follows^ :— " From every not-B is not- A he pro- 
duces Every A is B, thus : If it be possible, let this A be 
not-B, but every not-B is not-A, therefore this A is not-A, 
which is absurd : whence every A is B." Now Dc Morgan 
thinks that this proof is entirely needless, because common 
logic gives the inference without the use of any geo- 
metrical reasoning. I conceive however that logic gives 
the inference only by an indirect process. De Morgan 
claims " to see identity in Every A is B and every not-B 
is not-A, by a process of thought prior to syllogism. 
Whether prior to syllogism or not, I claim that it is not 
prior to the laws of thought and the process of substitutive 
inference, by which it may be undoubtedly demonstrated. 

Employmmt of the Contrapositive Proposition, 

We can frequently employ the contrapositive form of a 
proposition by the method of substitution ; and certain 
moods of the ancient syllogism, which we have hitherto 
passed over, may thus be satisfactorily comprehended in 
our system. Take for instance the following syUogism in 
the mood Camestres : — 

' PkUosophicnl Afajfttti/n}, Dec. 1 852 ; p. 437. 

" Whales are not true fish ; for they do not respire water, 
whereas true fish do respire water." 
Let us take 

A = whale 

B = true fish 

C = respiring water 
Tlie premises are of the forms 

A = Ac . ,\ 

B = BC I J 

Now, by the process of contraposition we obtain from 
the second premise 

and we can substitute this expression for c in Ci) ob- 
taining ^ ^' 

A = Ahc 
or "Whales are not true fish, not respiring water" 

The mood Cesare does not really differ'' from Camestres 
except in the order of the premises, and it could be ex- 
hibited in an exactly similar manner. 

The m(K)d Baroko gave much trouble to the old logicians 
who could not reduce it to the first figure in the same 
manner as the other moods, and were obliged to invent 
specially for it and for Bokardo, a method of Indirect 
Reduction closely analogous to the indirect proof of Euclid. 
Now these moods require no exceptional treatment in this 
system. Let us take as an instance of Baroko, the areu 
ment ^ 

AU heated solids give continuous spectra (i) 

Some nebula do not give continuous spectra (2) 

Therefore, some nebulae are not heated solids (^) 

Treating the little word some as an indeterminate adiec- 

tive of selection to which we assign a symbol like any 

other adjective, let ^ 

A = some 

B = nebulsB 

C = giving continuous specti-a 
rp, ^ = heated solids 

A he premises then become 

D = DC (I) 

XT . AB = ABc (2) 

tl^I ^^'^ ^'^ "^^ ""^^"^ ^y *^^« ^°^^«^fc method the cou. 
trapositive proposition 

» / 

I i; 

f ^'i 





c = cd 
and if we substitute this expression for c in (2) we have 

AB = ABcd 
the full meaning of which is that " some nebulae do not 
give continuous spectra and are not heated solids." 

We ini^lit similarly apply the contrapositive in many 
other instances. Take the argument, " All fixed stars are 
self-luminous ; but some of the heavenly bodies are not 
self-luminous, and are therefore not fixed stars." Taking 
our terms 

A = fixed stars 
B = self-luminous 
C = some 

D = heavenly bodies 
we have the premises 

A = AB, (i) 

CD = bCD (2) 

Now from (i) we can draw the contrapositive 

& = aJ 
and substituting this expression for h in (2) we obtain 

CD = abCD 
which expresses the conclusion of the ai-gument that some 
heavenly bodies are not fixed stars. 

Contrapositive of a Simple Identity, 

The reader should carefully note that when we apply 
the process of Indirect Inference to a simple identity of 
the form 

A = B 
we may obtain further results. If we wish to know what 
is the term not-B, we have as before, by the I^aw of Duality, 

h = Ah •!• ah 
and substituting for A we obtain 

h =^W) \ah = ah. 
But we may now also draw a second contrapositive ; for 
we have 

a = aB •!• ah, 
and substituting for B its equivalent A we have 

a = a A \ah == ah. 
Hence from the single identity A = B we can draw 
the two propositions 


a^ ah 

b = ab, 
and observing that these propositions have a common term 
ab we can make a new substitution, getting 

a = 5. 
This result is in strict accordance with the fundamental 
principles of inference, and it may be a question whether 
it IS not a self-evident result, independent of the steps of 
deduction by which we have reached it For where two 
classes are coincident like A and B, whatever is true of 
the one is true of the other ; what is excluded from the one 
must be excluded from the other similarly. Now as a 
bears to A exactly the same relation that h bears to B the 
identity of either pair follows from the identity of the 
other pair. In every identity, equality, or simUarity, we 
may argue from the negative of the one side to the nec^a- 
tive of the other. Thus at ordinary temperatures ^^ 

Mercury = liquid-metal, 
hence obviously 

• Not-mercury = not liquid-metal ; 
or since 

Sirius = brightest fixed star, 

it follows that whatever star is not the brightest is not 

fc>irius, and vice versd. Every correct dcfniition is of the 

lorm A = B, and may often require to be applied in the 
eqmvalent negative form. 

Let us take as an illustration of the mode of usin^r this 

result the argument following : ° 

Vowels are letters which can be sounded alone, (i) 
The letter w cannot be sounded alone ; ' (2) 

Therefore the letter lo is not a vowel. (3) 

Here we have a definition (i), and a comparison of a 
thing with that definition (2), leading to exclusion of the 
tiling from the class defined. 
Taking the terms 
A = vowel, 

B = letter which can be sounded alone, 
C = letter w, 
the premises are plainly of the forms 

A=B, (,j 

C = 6C. (2) 





H '1 


" ' 





Now by the Indirect method we obtain from (i) the 

6 = rt. 
and inserting in (2) the equivalent for 6 wo have 

C = aC, (3) 

IT " the letter w is not a vowel." 

Miscellaneous Examples of the Method. 

We can apply the Indirect Method of Inference however 
many may be the terms involved or the premises con- 
taining those terms. As the working of the method is 
best learnt from examples, I will take a case of two 
premises forming the syllogism Barbara : thus 

Iron is metal (i) 

Metal is element (2) 

If we want to ascertain what inference is possible concern- 
ing the term Iron, we develop the term by the Law of 
Duality. Iron must be either metal or not-metal; iron 
which is metal must be either element or not-element ; 
and similarly iron which is not-metal must be either 
element or not- element. There are then altogether four 
alternatives among which tlie description of iron must be 
contained ; thus 

Iron, metal, element, (a) 

Iron, metal, not-element, {fi) 

Iron, not-metal, element, (7) 

Iron, not-metal, not-element. (3) 

Our lirst premise informs us that iron is a metal, and if 
we substitute this description in (7) and (8) we shall have 
self-contradictory combinations. Our second premise like- 
wise informs us that metal is element, and applying this 
description to (yS) we again have self-contradiction, so that 
there renaains only (a) as a description of iron — our 
inference is 

Iron = iron, metal, element. 
To represent this process of reasoning in general symbok 

A = iron 
B = metal 
C = element, 
The premises of the pi-oblem take the forma 



A = AB (I) 

B = BO. (2) 

By the Law of Duality we have 

A = AB f Aft (5) 

A = AC .|- Ac. (4) 

Now, if we insert for A in the second side of (3) its 

description in (4), we obtain what I shall call the develop- 

merU of A with respect to B and C, namely 

A = ABC .| ABc .|. A6C -I- khc. (5) 

Wlierever the letters A or B appear in the second side of 

(5) substitute their equivalents given in (i) and (2), and 

the results stated at full length ai-e 

A = ABC I- ABCc -|. ABftC I- Al^Cc. 

The last three alternatives break the Law of Contradiction, 

so that 

A = ABC I- o I- o •!• o = ABC. 
This conclusion is, indeed, no more than we could obtain 
by the direct process of substitution, that is by substituting 
for B m (1), its description in (2) as in p. 55 ; it is the 
characteristic of the Indirect process that it gives all 
possible logical conclusions, both those which we have 
previously obtained, and an immense number of others or 
which the ancient logic took little or no account. From 
the same premises, for instance, we can obtain a description 
of the class not-element or c. By the Law of Duality we can 
develop c into four alternatives, thus 

c = ABc I- Abe I- aBc I- abc. 
If we substitute for A and B as before, we get 

c = ABCc j. ABbc I- aBCc .;• abc, 
and, stnking out the terms which break the Law of 
Contradiction, there remains 

c = abc, 
or what is not element is also not iron and not metal 
i ins Indirect Method of Inference thus furnishes a 
complete solution of the following ^Tohhm— Given any 
number of logical pi^emises or conditions, required the 
aescnptum of any class of objects, or of any tmn, as 
governed by tliose conditions. 

1 he steps of the process of inference may thus be 
concisely stated^ 

nf '*u^^ ^^^ ^^ ^^ Duality develop the utmost number 
01 alternatives which may exist in the description of the 






■/ ! 


* / 

required class or term as regards the terma involved in the 

2. For each term, in these alternatives substitute its 
description as given in the premises. 

3. Strike out every alternative which is then found to 
break the Law of Contradiction. 

4. The remaining terms may be equated to the term in 
question as the desired description. 

Mr. VemCs Problem, 

The need of some logical method more powerful and 
comprehensive than the old logic of Aristotle is strikingly 
illustrated by Mr. Venn in his most interesting and able 
article on Boole's logic* An easy example, originallv got, 
as he says, by the aid of my method as simply described 
in the Elementary Lessons in Logic, was proposed in 
examination and lecture-rooms to some hundred and fifty 
students as a problem in ordinary logic. It was answered 
by, at most, five or six of them. It was afterwaixls set, 
as an example on Boole's method, to a small class who 
had attended a few lectures on the nature of these 
symbolic methods. It was readily answered by half Or 
more of their number. 

The problem was as follows :— " The members of a board 
were all of them either bondholders, or shareholders, but 
not both ; and the bondholders as it happened, were all on 
the board. What conclusion can be drawn ? " The con- 
clusion wanted is, "No shareholders are bondholders." 
Now, as Mr. Venn says, nothing can look simpler than the 
following reasoning, wJien stated :—'* There can be no 
bondholders who are shareholders ; for if there were they 
must be either on the board, or off it. But they are not 
on it, by the first of the given statements ; nor off it, by 
the second." Yet from the want of any systematic mode 
of treating such a question only five or six of some 
hundred and fifty students could succeed in so simple a 

»^»nd; a Quarterly Review of Psychology I^n4 PhUoeophy : 
October, 1876, vol. i. p. 487. ^ *^ ^ ' 


By symbolic statement the problem is instantly solved. 

A = member of board 

B = bondholder 

= shareholder 
the premises are evidently 

A = ABc j. A6C 

B = AB. 
The class C or shareholders may in respect of A and B be 
developed into four alternatives, 

C = ABC .|. AbG I- aBC \ ahG. 
But substituting for A in the first and for B in the third 
alternative we get 

C = ABCc .|. AB5C .|- A^C |. aABG I- obQ. 
The first, second, and fourth alternatives in the above are 
self-contradictory combinations, and only the^e; strikin*^ 
them out there remain ** 

C = AhQ \ ahG = JC. 
the required answer. This symbolic reasoning is, I believe, 
the exact equivalent of Mr. Venn's reasoning, and I do 
not believe that the result can be attained in a simpler 
manner. Mr. Venn adds that he could adduce other 
similar instances, that is, instances showing the necessity 
of a better logical method. 

Abbreviation of the Process, 

Before proceeding to further illustrations of the use of 
this method, I must point out how much its practical 
employment can be simplified, and how much more easy 
It is than would appear from the description. When we 
want to effect at aU a thorough solution of a locrical 
problem it is best to form, in the first place, a complete 
series of all the combinations of terms involved in it If 
there be two terms A and B, the utmost variety of 
combinations in which they can appear are 

AB aB 

llie term A appears in the first and second ; B in the first 
and third ; a in the thii-d and fourth ; and b in the second 
and fourth. Now if we have any premise, say 

A = B, 






1 ,1 




we must ascertain which of these combinations will be 
rendered self-contradictory by substitution; tlie second 
and third will have to be struck out, and there will remain 
only AB 

Hence we draw the following inferences 

A = AB, B = AB, a^ab, h ^ ah. 
Exactly the same method must be followed when a 
question involves a greater number of terms. Thus by the 
Law of Duality the three terms A, B, C, give rise to eight 
conceivable combinations, namely 

ABC (a) aBC (e) 

ABc (fi) aWc (f) 

A6C (7) ahO irj) 

Ahc (5) ahc. (0) 

The development of the term A is formed by the first four 

of these; for B we must select (a), (J3), (c), (f); C 

consists of (a), (7), (e), (17) ; h of (7), (8), (rj\ {0), and so on. 

Now if we want to investigate complet<}ly the meaning 

of the premises A = AB (i) 

B = BC (2) 

we examine each of the eight combinations as regards each 
premise; (7) and (3) are contradicted by (i), and (fi) and 
(f ) by (2), so that there remain only 

ABC (a) 

aBC (c) 

ahC (ff) 

ahc. • (0) 

To describe any term under the conditions of the premises 
(i) and (2), we have simply to draw out the proper com- 
binations from this list; thus, A is represented only by 
ABC, that is to say 

A = ABC, 
similarly c = ahc. 

For B we have two alternatives thus stated, 

B = ABC i aBC ; 
and for h we have 

h = ahC •{' ahc. 
When we have a problem involving four distinct terms 
we need to double the number of combinations, and as 
we add each new term the combinations become twii*^ 
as numerous. Thus 

produce four combinations 





A, B 
A, 1^, C, 
A, B, C, D 
A, B, C, D, E 
A, B, C, D, E, F 
and so on. 

I pix)pose to cfill any such series of combinations the 
Logtml Alp/iahet, It holds in logical science a position 
the importance of which cannot be exaggerated, and as 
we proceed from logical to mathematical considerations it 
wiU become apparent that there is a close connection 
between these combinations and the fundamental theorems 
ot mathematical science. For the convenience of the 
reader who may wisli to employ the Alpliahet in logical 
questions, I liave had printed on the next page a complete 
senes of the combinations up to those of six terms At 
the very conimencenient, in the first column, is placed a 
smgle letter X. which might seem to be superfluous. This 
letter serves to denote that it is always some higher class 
which IS divided up. Thus the combination AB really 
means ABX, or that part of some larger class, say X 
which has the qualities of A and B present. The letter 
A IS omitted in the greater part of the table merely for the 
sake of brevity and clearness. In a later chapter on Com- 
binations It will become apparent that the introduction of 
tins unit class is requisite in order to complete the 
analogy with the Arithmetical Triangle there described. 
I he reader ought to bear in mind that though the Logical 
Alphabet seems to give mere lists of combinations, these 
combinations are intended in every case to constitute the 
development of a term of a proposition. Thus the four 
combinations AB, A6, aB, ah really mean that any class X 
is described by the following proposition 

, X = XAB .|. XAb .|. XaB .|. Xah. 
If we select the A's, we obtain the foUowing proposition 
_,. ^ AX = XAB .|. XA6. 

ihus whatever group of combinations we treat must be 

n2Zf ^K.rL'^^ \^^^'' "^'-^^^ ^^^^^ ff^^ or 
umverse symbo ised in the term X ; but, bearing this in 

mind It 13 needless to complicate our formulae by always 

mtroducing the letter. All inference consists in passing 

»rom propositions to propositions, and combinations^^ ^ 






[cf a9. 

have no meaning. They are consequently to be regarded 
in all cases as forming parts of propositions. 

The Logical Alphabet. 









A B 




a JL 

A b 

A B e 

A B Cd 


«. B 

A bC 

A B e D 


• h 


A B ed 

A B C d< 

fc U O 

Aft C D 


a B e 

A b C d 

A Be D < 

« ft C 

A b e D 

A Bed E 

a b 9 

aDC D 
aB Cd 
a B e D 
a B e d 
o !» «" J> 
a ft d 
n ft c i> 
rt ft - d 

A U ed « 
A bC D« 
A b C d B 
Ab C d f 
Abe D B 
A b e D < 
A be d B 
A b e d « 
a n C 1) « 
aBC d B 
a U Cd « 
ABe D B 
II Jl e D < 
« Be d B 
A R e d e 
« ft C D B 
u ( < 

• ft Cd B 

• ft C d < 
abe D B 
a 6 r D < 
akt d ¥, 



A B C I) e P 
A B C D e/ 
A B C d E K 
A BC d E/ 
A BC de P 
A B C d e / 
A Be D E K 
ABe D E/ 
A Be De P 
A B c D e/ 
A Bed E P 
A Be d E/ 
A B ede P 
A B e d e / 
AbC D E/ 
AbC De P 
A bC D «/ 
AbCd E P 
Ab Cd e F 
AbC dt f 
AbcD E P 
A be D E/ 
A beD « P 
A b e D « / 
Abed E P 
Abed K / 
A b c d e P 
A b e d < / 
a B C D E P 
aBO De P 
a B D e / 
aBC d BP 
a B Cd B/ 
a BCd e P 
a R C d e / 
aBe DE P 
•« Be D B/ 
a B e D « P 
« B e D e / 
a B e d E K 
IT B e d K / 
a B e d e P 
a B e d e / 
a b C D E F 
■ bC D B/ 
nb C D • P 
« b C D e / 
abC d K P 
A b C d E/ 
a b C d e P 
a b C d « / 
a B e D E P 
a b e D B / 
« b e D e P 
a b e D < / 
a b e d E P 
^h e d E / 
aft e d « P 
a ft e d « / 

"1 J:?12™raV»ETH0DOTJNmEl.C£ „ 

of such combinations whe?, ,,!flf • I ^ *u " """"^^ 

?ilrs TdF^ =^ ats^^ :^ 

deep lo^cal importance 3 ^^ fTTh^TT^ '''^ 
the symbol of identity and hanno'nv l,„ J^ -1% "^^ 
number two as the ori.-in nf 1^ ^' ''^ described the 
dive«ity. division and^pa^tfo''^^' "' '^'. ^y""''"' »' 
the reWy,, was also reSd hv . • '^ """''^'' *■«»"•' »•■ 
elements o/ ;xistence! K iK"ted Zl "' *'^ ''i""^' 
virtue whence come all combSon, T„ generating 
golden verses ascribed to TvZir! K ^""^ °^ '''« 
pupa to be virtuous : ' 'J^hagorsH, he conjures his 

"tL P? "h" °^"'P' ^*« ■*'<'«'• "IX"' tlie Mind 
rA« «,„r. tl,e fount of Nature', en.lleas S." 

Now four and the higher nowpr* nf .i...i-* j 
in this logical svstem th7n,l^7 i '*'''y ''•' "^Present 
can be SnS in fhl T^*^ '''l?"''*"*'«"n«^'Wcl. 
The fol W« S^Pythatoras' mfv K '"S''?^ restrictions, 
master's doctrines in raSL™ ^-, *''* '^'^''^^ "'«*'• 
but in many w)i,faT!!T^,'^"'* superstitious notions, 
basis in logiiEi;'!,^ •^^*""«« ^eem to have some 

Tkfi Logical State. 

sigli^i'carcrand '"tiiuvT T^f^^'''^ ^^e ext:.me • 
indirect proems Tllj! *? ^^^^ Alphabet the 

repetition'^^a few n^^^^^ "^T"* '^^''^ ^ the 
selection, and e 'mina£ nf^"* 'T ^'^ "'"s^ification. 
deduction, even Tn the „1,"°"*'*<^,«''^"««- I^ieal 

becomes k mat er of ZrTrLT^^T^ ^««««'»»«. 

mere routine, and the amount of 

' Wl..well, mi^ of iU Inductive SoUnce., vol. 1. p. „^ 

'J :• 

• r' 

■•- > ': 



labour required is the only impediment, when once the 
meaning of the premises is rendered clear. But the 
amount of labour is often found to be considerable. The 
mere writing down of sixty-four combinations of six 
letters each is no small task, and, if we had a problem of 
five premises, each of the sixty-four combinations would 
have to be examined in connection with each premise. 
The requisite comparison is often of a very tedious 
character, and considerable chance of error intervenes. 

I have given much attention, therefore, to lessening both 
the manual and mental labour of the process, and I shall 
describe several devices which may be adopted for saving 
trouble and risk of mistake. 

In the first place, as the same sets of combinations occur 
over and over again in different problems, we may avoid 
the labour of writing them out by having the sets of 
letters ready printed upon small sheets of \vriting-paper. 
It has also been suggested by a correspondent that, if any 
one series of combinations were marked upon the margin 
of a sheet of paper, and a slit cut between each pair of 
combinations, it would be easy to fold down any particular 
combination, and thus strike it out of view. The com- 
binations consistent with the premises would then remain 
in a broken series. This method answers sufficiently well 
for occasional use. 

A more convenient mode, however, is to have the series 
of letters shown on p. 94, engraved upon a common school 
writing slate, of such a size, that the letters may occupy 
only about a third of the space on the left hand side of 
the slate. The conditions of the problem can then be 
written down on the unoccupied part of the slate, and the 
proper series of combinations being chosen, the contra- 
dictory combinations can be struck out with the pencil. 
I have used a slate of this kind, which I call a Logical 
Slate, for more than twelve years, and it has saved me 
much trouble. It is hardly possible to apply this 
process to problems of more than six terms, owing to 
the large number of combinations which would require 


Abstraction of Indifferent Circumstances, 

There is a simple but highly important process of 
inference which enables us to absti-act, eliminate or dis- 
regard all circumstances indifferently present and absent 
Thus if I were to state that " a triangle is a three-sided 
rectilinear figure, either large or not lai^e," these two 
alternatives would be superfluous, because, by the Law of 
Duality, I know that everything must be either large or 
notlai^e. To add the qualification gives no new know- 
ledge, since the existence of the two alternatives wiU be 
understood in the absence of any information to the 
contrary. Accordingly, when two alternatives differ only 
as regards a single component term which is positive in 
one and negative in the other, we may reduce them to one 
term by striking out their indifferent part. It is really a 
process of substitution which enables us to do tliis • for 
having any proposition of Ihe form ' 

A = ABC J. ABc, (I) 

we know by the Law of Duality that 

, , AB=ABC .|. ABc. (2) 

As the second member of this is identical witd the s^ond 
member of (i) we may substitute, obtaining 

A = AB. 
This process of reducing .useless alternatives may bo 
applied again and again ; for it is plain that 

A = AB (CD .|. Qd .|- cD .|. cd) 
cfimmunicates no more information than that A is B 
Abstraction of indifferent terms is in fact the converse 
process to that of development described in p. 89; and 
It IS one of the most important operations in the whole 
sphere of reasoning. 
The reader should observe that in the proposition 

AC = BC 
we cannot abstract C and infer 

but from 

AC ^. Ac = BC 1- Be 
we may abstract all reference to the term C. 

It ought to be carefully remarked, however, that alter- 
natives which seem to be without meaning often imply 
important knowledge. Thus if I say that " a triangle w a 


; . 











three-sided rectilinear figure, with or without three equal 
angles," the last alternatives really express a property of 
triangles, namely, that some triangles have three equal 
angles, and some do not have them. If we put P =» 
" Some," meaning by the indefinite adjective " Some," one 
or more of the undefined properties of triangles with three 
equal angles, and take 
A = triangle 

B = three-sided rectilinear figure, 
C = with three equal angles, 
then the knowledge implied is expressed in the two 

joA = pBc, 
These may also be thrown into the form of one pro- 
position, namely, 

A = PBC I- pBc; 
but these alternatives cannot be reduced, and the propo- 
sition is quite different from 

A = BC I- Be 

Jllustrations of the Indirect Method. 

A great variety of arguments and logical problems 
might be introduced here to show the comprehensive 
character and powers of the Indirect Method. We can 
treat either a single premise or a series of premises. 

Take in the first place a simple definition, such as " a 
triangle is a three-sided rectilinear figure." Let 

A = triangle 
B =s three-sided 
C "= rectilinear figure ; 
then the definition is of the form 

A = BC. 
If we take the series of eight combinations of three 
letters in the Logical Alphabet (p. 94) and strike out 
those which are inconsistent with the definition, we have 
♦JiP following result : — ABC 


For the description of the class C we have 

C = ABC A' ahC, 
that is, " a rectilinear figure is either a triangle and three- 
sided, or not a triangle and not three-sided." 
For the class b we have 

h = ahC •(• abc. 
To the second side of this we may apply the profr<»ss of 
simplification by abstraction described in the last section • 
for by the Law of Duality * 

ah = abC •(• abc ; 
and as we have two propositions identical in the second 
side of each we may substitute, getting 

h — aby 
or what is not three-sided is not a triangle (wliether it be 
rectilinear or not). 

Second Example, 

H! S^ ^^^\^y ^^^ method the following ai^ument :— 
" Blende is not an elementary substance ; elementary 
substances are those which are undecomposable ; 
blende, therefore, \& decomposable." 
Taking our letters thus— 

A = blende, 
« B = elementary substance, 

C = undecomposable, 
the premises are of the forms 

A = AJ, (,) 

B = C. (2) 

No immediate substitution can be made ; but if we take 

lUe contrapositive of (2) (see p. ^6), namely 

i = c, /^\ 

we can substitute in (i) obtaining the conclusion 

mrnhf ^•^°'® result may bl obtained by taking the eight 

h^t^'^A'?^.''^ .^' ?' ^' ^^ *^^ Logical Alphabet; it wiU 
DC lound that only three combinations, namely 

Abe . 

are consistent with the premises, whence it results thai 

A = Abc, 

H 2 

? I 

r I 




or by the process of Ellipsis before described (p. 57) 

A = Ac. 

Third Example, 

As a somewhat more complex example I take the 
argument thus stated, one which could not be thrown into 
the syllogistic form : — 

" All metals except gold and silver are opaque ; there- 
fore what is not opaque is either gold or silver or 
is not-metal." 
There is more implied in this statement than is dis^ 
linctly asserted, the full meaning being as follows: 




All metals not gold or silver are opaque, 
Gold is not opaque but is a metal, 
Silver is not opaque but is a metal, 
Gold is not silver. 
Taking our letters thus — 

A = metal C = silver 

^ = gold J) = opaque, 

we may state the premises in the forms 

Abe = AhcD 
B = AB</ 
C = ACd 
B = Be. 
To obtain a complete solution of the question we Take 
the sixteen combinations of A, B, C, D, and striking out 
those which are inconsistent with the premises, there i-emain 

; abed. 

The expression for not-opaque things consists of the 
three combinations containing d, thus 

d==ABed •]• AbCd .|. abed, 
or rf " A<£ (Be .|. bC) + al)cd. 

In ordiuar}' language, what is not-opaque is either metal 
which is gold, and then not-silver, or silver and then not 
gold, or else it is not-metal and neither gold nor silver. 


Fourth Example. 

A good example for the illustration of the Indirect 
Method is to be found in De Morgan's Formal Logic (p 
123), the premises being substantially as follows :-~ 

From A follows B, and from follows D ; but B and D 
are inconsistent with each other ; therefore A and C are 

The meaning no doubt is that where A is, B will be 
found, or that every A is a B, and similarly eveiy C is a D • 
but B and D cannot occur together. The premises there- 
fore appear to be of the forms 

A = AB, (,) 

C = CD, 2 

B = Brf. ).{ 

On examining the series of sixteen combinations, only 
five are found to be consistent with the above condition? 

In these combinations the only A which appears is joined 
with C ^^""'^^'■^^ ^ ^^ J^'"^^ *^ a, or A is inconsistent 

Fifth Example. 

A more complex argument, also given by De Morgan i 
contjains five t^rms, and is as stated below, except that 
the letters are altered. » ^ * 

Every A is one only of the two B or C ; D is both B 

and C, except when B is E, and then it is 

neither ; therefore no A is D. 

The meaning of the above premises is difficult to 

interpret, but seems to be capable of expression in the 

foUowmg symbolic forms— 

PointenSi ^' ?l '^^ ^^ Professor Croom Robrt^on has 

mto T.i«^* ^^ ™^? ^^^ S^^'**^ *°^ ^^"^ premises out be thrown 
into a single propontion, D - D#BO f DKic iwrown 

I' I 











A = AB(; I- AbQ, (i) 

De = DeBC, (2) 

DE=DE^»c. (3) 

As five terms enter into these premises it is requisite to 
treat their thirty-two combinations, and it will be found 
that fourteen of them remain consistent with the premises 
namely ' 




If we examine the first four combinations, all of which 
contain A, we find that they none of them contain D ; or 
again, if we select those which contain D, we have only 
two, thus — 

D = aBCD(5 .[. alcDE. 
Hence it is clear that no A is D, and vice versd no D is A. 
We might draw many other conclusions from the same 
premises ; for instance — 

DE = ahcDE, 
or D and E never meet but in the absence of A, B, and C. 

Fallacies analysed by the Indirect Method. 
It has been sufficiently shown, perhaps, that we can by 
the Indirect Method of Inference extract the whole truth 
fron) a series of propositions, and exhibit it anew in any 
required form of conclusion. But it may also need to be 
shown by examples that so long as we follow conectly 
the almost mechanical rules of the method, we cannot fall 
into any of the fallacies or paralogisms which are often 
committed in ordinary discussion. Let us take the example 
of a fallacious argument, previously treated by the Method 
of Direct Inference (p. 62), 

Granite is not a sedimentaiy rock, (i 

Basalt is not a sedimentary rock, (21 

and let us ascertain whether any precise conclusion can be 
drawn concerning the relation of granite and basalt 
Taking as before 

A = granite, 

B = sedimentary rock, 

C =» basalt. 


the premises become A^ Ah (i) 

C = C^' (2) 

Of the eight conceivable combinations of A, B. 0. five 
agree with these conditions, namely 

AbG aBe 

Abe ahC 

Selecting the combinations which contain A, we find the 
description of granite to be 

A = A5C |. Abc=^Ab(G •(. c), 
that IS, granite is not a sedimentary rock, and is either 
basalt or not-basalt. If we want a description of basalt the 
answer is of like form 

C = A^C .|. abG = bC(A + a), 
that 18 basalt is not a sedimentary rock, and is either 
granite or not-granite. As it is ahready perfectly evident 
that basalt must be either granite or not, and vice versd 
the premises fail to give us any information on the point' 
that IS to say the Method of Indirect Inference saves us 
from falling into any fallacious conclusions. This 
example sufficiently iUustrates both the fallacy of 
Negative premises and that of Undistributed Middle of 
the old logic 

The faUacy called the Illicit Process of the Major Term 
IS also incapable of commission in following the rules of 
the method. Our example was (p. 65) 

All planets are subject to gravity, (i) 

Fixed stars are not planets. (2) 

The false conclusion is that " fixed stars are not subject to 
gravity." The terms are ^ 

A = planet 
B = fixed star 
- , ^_ C = subject to gravity. 

And the premises are A = AC, (i) 

-^ B = aB. (2) 

The combinations which remain uncontradicted on com- 
parison with these premises are 

AbG aBe 

aBG abG 

For fixed star we have the description 

B = aBC A- aBc 

t fmi 




ni^Ll ,. *i, **" *' ?"' * P''^"^'- ^"' i« «*« subject 
or not. as the case may be, to gravity." Here we have no 

conclusion concerning the connection of fixed stars and 
gravity. " 

The Logical Abaeut. 

_ The Indirect Method of Inference lias now been suffi 
ciently described, and a careful examina^on of ^ — 

Sfchief diSu'^''^''?? '"^°^r"." °"'y >««*<='*» ^J^tions 
ine chief difficulty of the method consists in the great 

number of combinations which may have to be exami^ 

not on^y may the requisite labour become form'dabTe but 

a considerable chance of mistake arises. I hive the^foS 

given much attention to modes of facilitating the work 

miharafT™ '''t^ ^"""^ the method t^ an alS 
mechanical form It soon appeared obvious that if the 
conceivable combinations of the Logical Alphabet for anv 

T^l^'fr- ^''^^ f being^printed^" fix'ed onTr 
^n^oKp^ PfP*' °' ='«•*• '^ere marked upon lieht 
ITi iP'r '? "/ 7'^' niechanical armngemeSts could 

St thr ;<. ^* ^*^"' "I comparison and rejection 

Si in fh- if ^^'^ "^i"^"*"' ^'''<='' I have found 
rXln nf w- Y"1"r""" ^""^ exhibiting the complete 
solution of logical problems. A minute description of the 
oo.«truction and use of the Abacus, together wiXfigures 
mlZT^^ ^'^^^.,heen given if my essay cE 
JAe bubdUutum of Simtiars.' and I will here cive onlv 
a general description. *» ' 

ho^l^S*' ^hacus «)nsists of a common school black- 
board placed in a sloping position and furnished with four 

of thTSte^^shT"^'^^ ^^«^- ^""^ cornhZtZ 
lUnfi if^ t?""" '" *he first four columns of the 
Logical Alphabet are printed in somewhat lame type 
80 that each letter is about an inch from the neTh W 

inst^ of being m horizontal lines as in p at. Each 
combumtion of letters is sepamtely fixed to the^urface of 

' I> SS-S* 81—86. 





a thin slip of wood one inch broad and about one-eighth 
inch thick Short steel pins are then driven in an inclined 
position into the wood. When a letter is a large capital 
representing a positive term, the pin is fixed in the upper 
part of its space ; when the letter is a small italic repre- 
senting a negative term, the pin is fixed in the lower part 
of the space. Now, if one of the series of combinations 
be ranged upon a ledge of the black-board, the sharp edge 
of a flat rule can be inserted beneath the pins belonging to 
xny one letter — say A, so that all the combinations marked 
A can be lifted out and placed upon a separate ledge. 
Thus we have represented the act of thought which 
separates the class A from what is not- A. The operation 
can be repeated ; out of the A's we can in like manner 
select those which are B s, obtaining the AB's ; and in like 
manner we may select any other classes such as the aB's, 
the ab*B, or the ahc*8. 

If now we take the series of eight combinations of the 
letters A, B, C, a, b, c, and wish to analyse the argument 
anciently called Barbara, having the premises 

A = AB (I) 

B = BC, (2) 

we proceed as follows — We raise the combinations marked 
a, leaving the A's behind; out of these A's we move to a 
lower ledge such as are 6's, and to the remaining AB's 
we join the a's which have been raised. The result is that 
we have divided all the combinations into two classes, 
namely, the Aft's which are incapable of existing consist- 
ently with premise (i), and the combinations which are 
consistent with the premise. Turning now to the second 
premise, we raise out of those which agree with (i) the i's, 
then we lower the IVs ; lastly we join the 5*s to the BC's. 
We now find our combinations arranged as below. 

























The lower line contains all the combinations which are 
inconsistent with either premise ; we have carried out in a 


; 1 




mechanical manner that exclusion of self-contradictories 
which was formerly done upon the slate or upon paper 
Accordingly, from the combinations remaining in the upper 
line we can draw any inference which the premises yield 
If we raise the A's we find only one, and that is C so 
that A must be C. If we select the c s we again find only 
one, which is a and also 3 ; thus we prove that not-C is 
not-A and not-B. 

When a disjunctive proposition occurs among the 
premises the requisite movements become rather more 
complicated. Take the disjunctive argument 

A is either B or C or D, 
A is not C and not D, 
Therefore A is B. 
The premises are represented accurately as follows :— 

A = AB t AC I- AD (I) 

A = Ad. /^\ 

As there are four terms, we choose the series of sixteen 
combinations and place them on the highest ledffe of the 
board but one. We raise the a's and out of the A's. which 
remain, we lower the b's. But we are not to reject all the 
A6 s as contradictory, because by the first premise A's 
may be either Bs or C's or D's. Accordingly out of the 
Ah 8 we must select the c% and out of these again the rf's 
80 that only Abed will remain to be rejected finally! 
Joining all the other fifteen combinations together aaain 
and proceeding to premise (2), we raise the a's and iSwer 
the AGs, and thus reject the combinations inconsistent 
with (2) ; similarly we reject the AD's which are incon- 
sistent with (3) It will be found that there remain, in 
addition to all the eight combinations containing a only 
one contiiining A, namely ' 

whence it is apparent that A must be B, the ordinary 
conclusion of the argument. 

In my "Substitution of Similars" (pp. 56—50) I have 
described the working upon the Abacus of two other 
logical problems, which it would be tedious to repeat in 
tins place. '^ 



The Logical Machine. 

Although the Logical Abacus considerably reduced the 
lalwur of using the Indirect Method, it was not free from 
the possibility of error. I thought moreover that it would 
affortl a conspicuous proof of the generality and power of 
the method if I could reduce it to a purely mechanical 
form. Logicians had long been accustomed to speak of 
Logic as an Organon or Instrument, and even Lord Bacon, 
while he rejected the old syllogistic logic, had insisted, in 
the second aphorism of his " New Instrument," that the 
mind required some kind of systematic aid. In the 
kindred science of mathematics mechanical assistance of 
one kind or another had long been employed. Orreries, 
globes, mechanical clocks, and such like instruments, 
are really aids to calculation and are of considerable 
antiquity. The Arithmetical Abacus is still in common 
use in Russia and China. The calculating machine of 
Pascal is more than two centuries old, having been con- 
structed in 1642-45. M. Thomas of Colmar manufactures 
an arithmetical machine on Pascal's principles which is 
employed by engineers and others who need frequently 
to multiply or divide. To Babbage and Scheutz is due 
the merit of embodying the Calculus of Differences in a 
machine, which thus became capable of calculating the 
most complicated tables of figures. It seemed strange 
that in the more intricate science of quantity mechanism 
should be applicable, whereas in the simple science of 
qualitative reasoning, the syllogism was only called an 
instrument by a figure of speech. It is true that Swift 
satirically described the Professors of Laputa as in pos- 
session of a thinking machine, and in 185 1 Mr. Alfred 
Smee actually proposed the construction of a Relational 
machine and a Differential machine, the first of which 
would be a mechanical dictionary and the second a mode 
of comparing ideas; but with these exceptions I have 
not yet met with so much as a suggestion of a reasoning 
machine. It may be added that Mr. Smee's designs, though 
highly ingenious, appear to be impracticable, and in . any 
case they do not attempt the performance of logical inference.^ 

* See hiB work called The Process of Thought adapted to Words and 
LanguagSy together with a Description of the Belationcd and Differ- 





The Logical Abacus soon suggested the notion of a 
Logical Machine, which, after two unsuccessful attempts 
I succeeded m constructing in a comparatively simple and 
effective form. The detaUs of the Logical Machine have 
been fuUy described by the aid of plates in the Philo- 
sophical Transactions,! and it would be needless to repeat 
the account of the somewhat intricate movements of the 
machine in this place. 

The general appearance of the machine is shown in a 
plate facing the title-page of this volume. It somewhat 
resembles a very small upright piano or organ, and has a 
keyboard containing- twenty-one keys. These keys are of 
two kinds, sixteen of them representing the terms or 
letters A a, B, 5, C, c, D, d, which have so often been 
employed in our logical notation. When letters occur on 
the left-hand side of a proposition, formerly called the 
subject each is represented by a key on the left-hand half 
of the keyboard ; but when they occur on the right-hand 
side, or as it used to be called the predicate of the pro- 
position the letter-keys on the right-hand side of the 
keyboard are the proper representatives. The five other 
keys may be called operation keys, to distinguish them 
trom the letter or term keys. They stand for the stops 
copula,, and disjunctive conjunctions of a proposition. 
The middle key of all is the copida, to be pressed when 
the verb is or the sign = is met. The key to the extreme 
nght-hand is caUed the Full Stop, because it should be 
pressed when a proposition is completed, in fact in the 
proper place of the full stop. The key to the extreme 
lelt-hand is used to terminate an ai^ument or to restore 
the machine to its initial condition ; it is called the Finis 
key. The last keys but one on the right and left com- 
plete the whole series, and represent the conjunction or in 
Its unexclusive meaning, or the sign H which I have 
employed, according as it occurs in the right or left hand 
side of the proposition. The whole keyboard is arranged 
as shown on the next page — 

mtialMaehinei, Alao PhUotophical Trantactioin, [1870] vol. 160. 
of Q^ B^al Socuijf, vol. xvui. p. 166, Jan. 20 djo. iyTolm, vol.1 


LefUhind ikl« of Proposition. 


Right-haud side of Proposition. 


















D d 


To work the machine it is only requisite to press the 
keys in succession as indicated by the letters and signs of 
a symbolical proposition. All the premises of an argu- 
ment are supposed to be reduced to the simple notation 
which has been employed in the previous pages. Taking 
then such a simple proposition as 

A = AB, 
we press the keys A (left), copula, A (right), B (right), and 
full stop. 

If there be a second premise, for instance 

B = BC, 
we press in like manner the keys — 

B (left), copula, B (right), C (right), full stop. 
The process is exactly the same however numerous the 
premises may be. When they are completed the operator 
will see indicated on the face of the machine the exact 
combinations of letters which are consistent with the 
premises according to the principles of thought. 
' As shown in the figure opposite the title-page, the 
machine exhibits in front a Logical Alphabet of sixteen 
combinations, exactly like that of the Abacus, except 
that the letters of each combination are separated by a 
certain interval. After the above problem has been 
worked upon the machine the Logical Alphabet will have 
been modified so as to present the following appearance — 




a\ a a a 

B B 1 



\b h b b 



C C « 1 c' 



0|4 1 


D d 

D d DJ d 







^ The operator will readily collect the various conclusions 
m the manner described in previous pages, as for in- 
stance that A is always C, that not-C is not-B and not- 
A ; and not-B is not-A but either C or not-C. The results 
are thus to be read off exactly as in the case of the 
ijogical Slate, or the Logical Abacus. 

Disjunctive propositions are to be treated in an exactly 
similar manner. Thus, to work the premises 

A = AB .|. AC i 

B + C = BD.|.CD, 
It IS only necessary to press in succession the keys 

T> A.^J^^9' ^P""^^' A ^"Sht), B, .|. , A,C, full stop. 
B aeft). .|. C, copula, B (right). D, + , C,D, full stop. 
Ihe combinations then remaining will be as follows 
ABCD aBCt) abd) 

ABcB aBcB abed. 

AbCD abCB 

On pressing the left-hand key A, aU the possible com- 
bmations which do not contain A will disappear, and the 
description of A may be gathered from what remain 
namely that it is always D. The full-stop key restores aU 
combinations consistent with the premises and any other 
selection may be made, as say not-D, which wUl be found 
to be always not-A, not-B, and not-C. 

At the end of every problem, when no further questions 
need be addressed to the machine, we press the Finis 
key, which has the effect of bringing into view the whole 
ot the conceivable combinations of the alphabet This 
key in fact obliterates the conditions impressed upon the 
machine by moving back into their ordinary places those 
combinations which had been rejected as inconsistent with 
the premises. Before beginning any new problem it is 
requisite to observe that the whole sixteen combinations 
are visible. After the Finis key has been used the machine 
represents a mmd endowed with powers of thought, but 
whoUy devoid of knowledge. It would not in that con- 
dition give any answer but such as would consist in the 
primary laws of thought themselves. But when any pro- 
position IS worked upon the keys, the machine analyses 
aiid digests the meaning of it and becomes charged with 
the knowledge embodied in that proposition. Accordingly 
It 18 able to return as an answer any description of a term 

or class so far as furnished by that proposition in accordance 
with the Laws of Thought. The machine is thus the em- 
bodiment of a true logical system. The combinations are 
classified, selected or rejected, just as they should be by a 
reasoning mind, so that at each step in a problem, the 
Logical Alphabet represents the proper condition of a mind 
exempt from mistake. It cannot be asserted indeed that 
the machine entirely supersedes the agency of conscious 
thought; mental labour is required in interpreting the 
meaning of grammatical expressions, and in con*ectly im- 
pressing that meaning on the machine ; it is further required 
in gathering the conclusion from the remaining combina- 
tions. Nevertheless the true process of logicsd inference 
is really accomplished in a purely mechanical manner. 

It is worthy of remark that the machine can detect any 
self-contradiction existing between the premises presented 
to it ; should the premises be self-contradictory it will be 
found that one or more of the letter-terms disappears 
entirely from the Logical Alphabet. Thus if we work the 
two propositions, A is B. and A is not-B, and then inquire 
for a description of A, the machine will refuse to give it 
by exhibiting no combination at all containing A. This 
result is in agreement with the law, which I have ex- 
plained, that every term must have its negative (p. 74). 
Accordingly, whenever any one of the letters A, B, C, D, a, 
6, c, d, wholly disappears from the alphabet, it may be 
safely inferred that some act of self-contradiction has been 

It ought to be carefully observed that the logical 
machine cannot receive a simple identity of the form 
A — B except in the double form of A = B and B = A. 
To work the proposition A = B, it is therefore necessary to 
press the keys — 

A (left), copula, B (rights full stop ; 
B (left), copula, A (right), full stop. 
The same double operation will be necessary whenever the 
proposition is not of the kind called a partial identity 
(p. 40). Thus AB = CD, AB = AC, A = B i C, A j- B 
= C .|. D, all require to be read from both ends separately. 

The proper rule for using the machine may in fact be 
given in the following way : — (i) Bead each proposition as 
it UandSf and play ihe corresponding keys : (2) Convert tJu 

1^1' I 






proposition and read atid play the keys again in the trans- 
posed order of the terms. So long as this rule is observed 
the true result must always be obtained. There can be no 
mistake. But it will be found that in the case of partial 
identities, and some other similar forms of propositions, 
the transposed reading has no effect upon the combinations 
of the Ix^ical Alphabet. One reading is in sucli cases all 
that is practically needful After some experience has 
been gained in the use of the machine, the worker naturally 
saves himself the trouble of the second reading when 

It is no doubt a remarkable fact that a simple identity 
cannot be impressed upon the machine except in the form 
of two partial identities, and this may be thought by some 
logicians to militate against the equational mode of repre- 
senting propositions. 

Before leaving the subject I may remark that these 
mechanical devices are not likely to possess much 
practical utility. We do not require in common life to be 
constantly solving complex logical questions. Even in 
mathematical calculation the ordinary rules of arithmetic 
are generally sufficient, and a calculating machine can only 
be used with advantage in peculiar cases. But the machine 
and abacus have nevertheless two important uses. 

In the first place I hope that the time is not very far 
distant when the predominance of the ancient Aristotelian 
Logic will be a matter of history only, and when the 
teaching of logic will be placed on a footing more worthy 
of its supreme importanca It will then be found that the 
solution of logical questions is an exercise of mind at least 
as valuable and necessary as mathematical calculation. I 
believe that these mechanical devices, or something of the 
same kind, will then become useful for exhibiting to a 
class of students a clear and visible analysis of logical 
problems of any degree of complexity, the nature of each 
step being rendered plain to the eyes of the students. I 
often used the machine or abacus for this purpose in 
my class lectures whUe I was Professor of Logic at 
Owens College. 

Secondly, the more immediate importance of the machine 
seems to consist in the unquestionable proof which it 
affords that correct views of the fundamental principles of 

reasoning have now been attained, although they were 
unknown to Aristotle and his followers. The time must 
come when the inevitable results of the admirable 
investigations of the late Dr. Boole must be recognised 
at their true value, and the plain and palpable form in 
which the machine piesents those results will, I hope, hasten 
the time. Undoubtedly Boole's life marks an era in the 
science of human reason. It may seem stranj^e that it had 
remained for him first to set forth in its full extent the 
problem of logic, but I am not aware that anyone before 
him had treated logic as a symbolic method for evolving 
from any premises the description of any class whatsoever 
as defined by those premises. In spite of several serious 
errors into which he fell, it will probably be allowed that 
Boole discovered the true and general form of logic, and 
put the science substantially into the form which it must 
hold for evermore. He thus effected a reform with which 
there is hardly anything comparable in the history of logic 
between his time and the remote age of Aristotle. 

Nevertheless, Boole's quasi* mathematical system could 
hardly be regarded as a final and unexceptionable solution 
of the problem. Not only did it require the manipulation 
of mathematical symbols in a very intricate and perplexing 
manner, but the results when obtained were devoid of 
demonstrative force, because they turned upon the employ- 
ment of unintelligible symbols, acquiring meaning only by 
analogy. I have also pointed out that he imported into 
his system a condition concerning the exclusive nature of 
alternatives (p. 70), which is not necessarily true of logical 
terms. I shall have to show in the next chapter that logic 
is really the basis of the whole science of mathematical 
reasoning, so that Boole inverted the true order of proof 
when he proposed to infer logical truths by algebraic 
processes. It is wonderful evidence of his mental power 
that by methods fundamentally false he should have 
succeeded in reaching true conclusions and widening the 
sphere of reason. 

The mechanical performance of logical inference affords 
a demonstration both of the truth of Boole's results and 
of the mistaken nature of his mode of deducing them. 
Conclusions which he coiild obtain only by pages of intri- 
cate calculation, ai-e exhibited by the machine after one or 







two minutes of manipulation. And not only are those 
conclusions easily reached, but they are demonstratively 
true, because every step of the process involves nothing 
more obscure than the three fundamental Laws of Thought 

The Order of Premises, 

Before quitting the subject of deductive reasomng, I 
may remark that the order in which the premises of an 
argument are placed is a matter of logical indifference. 
Much discussion has taken place at various times con- 
cerning the arrangement of the premises of a syllogism ; 
and it has been generally held, in accordance with the 
opinion of Aristotle, that the so-called major premise, 
containing the major term, or the predicate of the con- 
clusion, should stand first. This distinction however falls 
to the ground in our system, since the proposition is 
reduced to an identical form, in which there is no distinc- 
tion of subject and predicate. In a strictly logical pomt 
of view the order of statement is wholly devoid of 
significance. The premises are simultaneously coexistent, 
and are not related to each other according to the properties 
of space and time. Just as the qualities of the same 
object are neither before nor after each other in nature 
(p. 33), and are only thought of in some one order owing 
to the' limited capacity of mind, so the premises of an 
argument are neither before nor after each other, and are 
only thought of in succession because the mind cannot 
grasp many ideas at once. The combinations of the 
logical alphabet are exactly the same in whatever order 
the premises be treated on the logical slate or machine. 
Some difference may doubtless exist as regards convenience 
to human memory. -The mind may take in the results 
of an argument more easily in one mode of statement 
than another, although there is no real difference in the 
logical results. But in this point of view I think that 
Aristotle and the old logicians were clearly wrong. It is 
more easy to gather the conclusion that " all A's are C's ' 
from " all A's are B's and all B's are C's," than from the 
same propositions in inverted order, " all B's are C's and 
all A's ai-e B's. 

The Equivalence of Propositions 

One great advantage which arises from the study of 
this Indirect Method of Inference consists in the clear 
notion which we gain of the Equivalence of Propositions. 
The older logicians showed how from certain simple 
premises we might draw an inference, but they failed to 
point out whether that inference contained the whole, or 
only a part, of the information embodied in the premises. 
Any one proposition or group of propositions may be 
classed with respect to another proposition or qroup of 
propositions, as 

1. Equivalent, 

2. InfeiTible, 

3. Consistent, 

4. Contradictory. 

Taking the proposition " All men are mortals " as tlic 
original, then "All immortals are not men" is its equiva- 
lent ; " Some mortals are men " is infenible, or capable of 
inference, but is not equivalent ; ** All uot-men are not 
mortals" cannot be inferred, but is consistent, that is, 
may be true at the same time ; " All men are immortals " 
is of course contradictory. 

One sufficient test of equivalence is capability of mutual 
inference. Thus from 

All electrics = all non-conductors, 
I can infer 

All non-electrics = all conductors, 
and vice versd from the latter I can pass back to the 
former. In short, A = B is equivalent to a = b. Again, 
from the union of the two propositions, A = AB and 
B = AB, I get A = B, and from this I might as easily 
deduce the two with which I started. In this case one 
proposition is equivalent to two other propositions. There 
are in fact no less than four modes in which we may 
express the identity of two classes A and B, namely, 


A-B a = J B = Ab} 

The Indirect Method 01 Inference furnishes a universal 
and clear criterion as to the i-elationship of propositions. 
The import of a statement is always to be measured by 

I 2 


a = ab\ 
b = abf 




the combinations of terras which it destroys. Hence two 
propositions are equivalent when they remove the same 
combinations from the Logical Alphabet, and neither more 
nor less. A proposition is inferrible but not equivalent to 
another when it removes some but not all the combinations 
which the other removes, and none except what this 
other removes. Again, propositions are consistent provided 
that they jointly allow each term and the negative of 
each term to remain somewhere in the Logical Alphabet. 
If after all the combinations inconsistent with two propo- 
sitions are struck out, there still appears each of the letters 
A, a, B, h, C, c, D, rf, which were there before, then no 
inconsistency between the propositions exists, although 
they may not be equivalent or even inferrible. Finally, 
contradictory propositions are those which taken together 
remove any one or more letter-terms from the Logical 

What is true of single propositions applies also to groups 
of propositions, however large or complicated ; that is to 
say, one group may be equivalent, inferrible, consistent, 
or contradictory as regards another, and we may similarly 
compare one proposition with a group of propositions. 

To give in this place illustrations of all the four kinds 
of relation would require much space : as the examples 
given in previous sections or chapters may serve more or 
less to explain the relations of inference, consistency, and 
contradiction, I will only add a few instances of equivalent 
propositions or groups. 

In the following list each proposition or group of pro- 
j)Ositions is exactly equivalent in meaning to the corre- 
sponding one in the other column, and the truth of this 
statement may be tested by working out the combinations 
of the alphabet, which ought to be found exactly the same 
in the case of each pair of equivalents. 

A — b ... 

A = BC. . . 

A = AB.|.AC. 
A + B = C.|D. . 
A + c = B -I- (i . . 

A =» ABc I- A5C 


B = aB 

a = B 

a = h'\'C 

h^ab-l' AM) 
ah ^ cd 
aC = 5D 

A = AB I- AC 
AB«^ ABc 


A = B) / A = B 

B = Cj • • • \A = Q 

A = AB ) I A = AC 

B = BC j • • • t B = A I- aBC 

Although in these and many other cases the equivalents 

of certain propositions can readily be given, yet I believe 

that no uniform and infallible process can be pointed out 

by which the exact equivalents of premises can be 

ascertained. Ordinary deductive inference usually gives 

us only a portion of the contained information. It is 

true that the combinations consistent with a set of 

premises may always be thrown into the form of a 

proposition which must be logically equivalent to those 

premises ; but the difficulty consists in detecting the other 

forms of propositions which will be equivalent to the 

premises. The task is here of a different character from 

any which we have yet attempted. It is in reality an 

inverse process, and is just as much more troublesome and 

uncertain than the direct process, as seeking is compared 

with hiding. Not only may several different answers 

equally apply, but there is no method of discovering any 

of those answers except by repeated trial. The problem 

which we have here met is really that of induction, the 

inverse of deduction ; and, as I shall soon show, induction 

is always tentative, and, unless conducted with peculiar 

skill and insight, must be exceedingly laborious in cases 

of complexity. 

De Morgan was unfoitunately led by this equivalence of 
propositions into the most serious error of his ingenious 
system of Logic. He held that because the proposition 
" All A's are all B's," is but another expression for the 
two propositions " All A's are B*s " and " All B's are A's, 
it must be a composite and not really an elementary form 
of proposition.* But on taking a genei*al view of the 
equivalence of propositions such an objection seems to 
have no weight. Logicians have, with few exceptions, 
persistently upheld the original error of Aristotle in 
rejecting from their science the one simple relation of 
identity on which all more complex logical relations must 
really rest 

• Syllahus of a proposed syftem of Logic, §§ 57, 121, &c F(jfnnm 
Logic, p. 66^ 








TJie Nature of Inference, 


The question, What is Infereoce ? is involved, even to 
the present day, in as much uncertainty as that ancient 
question, What is Truth ? I shall in more than one part 
of this work endeavour to show that inference never does 
more than explicate, unfold, or develop the information 
contained in certain premises or facts. Neither in deduc- 
tive nor inductive i-easoning can we add a tittle to our 
implicit knowledge, which is like that contained in an 
unread book or a sealed letter. Sir W. Hamilton has well 
said, 'I Reasoning is the showing out explicitly that a 
proposition not granted or supposed, is implicitly contained 
in something dififerent, which is granted or supposed/' ^ 
^Professor Bowen has explained « with much clearness 
that the conclusion of an argument states explicitly what is 
virtually or implicitly thought. " The process of reasoning 
IS not so much a mode of evolving a new truth, as it is of 
establishing or proving an old one, by showing how much 
was admitted in the concession of the two premises taken 
together." It is true that the whole meaning of these 
statements rests upon that of such words as " explicit " 
* implicit," " virtual." That is implicit which is wrapped 
up, and we render it explicit when we unfold it. Just as 
the conception of a circle involves a hundred important 
geometrical properties, all following from what we know, 
if we have acuteness to unfold the results, so every fact 
and statement involves more meaning than seems at first 
sight. Reasoning explicates or brings to conscious posses- 
sion what was before unconscious. It does not create, nor 
does it destroy, but it transmutes and throws the same 
matter into a new form. 

The difficult question still remains, Wliere does novelty 
of form begin ? Is it a case of inference when we pass 
from " Sincerity is the parent of truth " to " The parent of 
truth IS smcerity ?" The old logicians would have called 
this change conversion, one case of immediate inference. But 
as all identity is necessarily reciprocal, and the very 
meanmg of such a proposition is that the two terms aro 

* Lectures on Metaphysics, vol. iv. p. 369. 

» Bowen, TreaHse on Logic, Ciuubridge, U.S., 1866 ; p. 362. 

identical in their signification, I fail to see any differencje 
between the statements whatever. As well might we say 
that X = 7/ and y = x are different equations. 

Another point of difficulty is to decide when a change 
is merely grammatical and when it involves a real logical 
transformation. Between a table of wood and a wooden 
table there is no logical difference (p. 31), the adjective 
being merely a convenient substitute for the prepositional 
phrase. But it is uncertain to my mind whether the 
change from " All men are mortal " to " No men are not 
nioital" is purely grammatical. Logical change may 
perhaps be best described as consisting in the determination 
of a relation between certain classes of objects from a 
relation between certain other classes. Thus I consider 
it a truly logical inference when we pass from " All men 
are mortal" to "AH immortals are not-men," because the 
clas.sos immortals and not-men are different from mortals 
and men, and yet the propositions contain at the bottom the 
very same truth, as shown in the combinations of the 
Logical Alphabet. 

The passage from the qualitacive to the quantitative 
mode of expressing a proposition is another kind of change 
which we must discriminate from true logical inference. 
We state the same truth when we say that "mortality 
belongs to all men," as when we assert that " all men arc 
mortals." Here we do not pass from class to class, but 
from one kind of term, the abstract, to another kind, the 
concrete. But inference probably enters when we pass 
from either of the above propositions to the assertion that 
the class of immortal men is zero, or contains no objects. 

It is of course a question of words to what processes we 
shall or shall not apply the name " inference," and I have 
no wish to continue the trifling discussions which have 
already taken place upon the subject. What we need to 
do is to define accurately the sense in which we use the 
word "inference," and to distinguish the relation of in- 
ferrible propositions from other possible relations. It 
seems to be sufficient to recognise four modes in which 
two apparently different propositions may be related. 
Thus two propositions may be — 

I. Tautologons or identical, involving the same relation 
between the same terms and classes, and only differing in 



khe order of statement ; thus " Victoria is the Queen of 
England " is tautologuus with " The Queen of England is 
Victoria." ° 

2. Grammatically/ related, when the classes or objects 
are the same and similarly related, and the only diflference 
13 in the words ; thus " Victoria is the Queen of Enoland " 
18 grammatically equivalent to "Victoria is England's 
Queen." ° 

3. Equivalents in qualitative and quantitative form the 
classes being the same, but viewed in a diflferent manner. 

4- Logically inferrible, one from the other, or it may be 
equivalent, when the classes and relations are dififerent but 
involve the same knowledge of the possible combinations 




We enter in this chapter upon the second great de- 
partment of logical method, that of Induction or the 
Inference of general from particular truths. It cannot 
be said that the Inductive process is of greater importance 
than the Deductive process already considered, because the 
latter process is absolutely essential to the existence of 
the former. Each is the complement and counterpart of 
the other. The principles of thought and existence which 
underlie them are at the bottom the same, just as subtrac- 
tion of numbei-s necessarily rests upon the same principles 
as addition. Induction is, in fact, the inverse operation 
of deduction, and cannot be conceived to exist without 
the corresponding operation, so that the question of re- 
lative importance cannot arise. Who thinks of asking 
whether addition or subtraction is the more important 
process in arithmetic? But at the same time much 
difference in difficulty may exist between a direct and 
inverse operation; the integral calculus, for instance, is 
infinitely more difficult than the differential calculus of 
which it is the inverse. Similarly, it must be allowed 
that inductive investigations are of a far higher degree of 
difficulty and complexity than any questions of deduction ; 
and it is this fact no doubt which led some logicians, such 
as Francis Bacon, Locke, and J. S. Mill, to erroneous^ 
opinions concerning the exclusive importance of induction. 
Hitherto we have been engaged in considering how from 
certain conditions, laws, or identities governing the com- 
binations of qualities, we may deduce the nature of the 



[en A p. 

combinations agreeing with those conditions. Our work 
has been to unfold the results of what is contained in any 
statements, and the process has been one of Synthesis. 
The terms or combinations of which the character has 
been determined have usually, though by no means always, 
involved more qualities, and therefore, by the relation of 
extension and intension, fewer objects than the terms in 
which they were described. The truths inferred were thus 
usually less general than the truths from wliich they were 

In induction all is inverted. The truths to be ascer- 
tained are more general than the data from which they 
are drawn. The process by which they are reached is 
analytical^ and consists in separating the complex com- 
binations in which natural phenomena are presented to 
us, and determining the relations of separate qualities. 
Given events obeying certain unknown laws, we have to 
discover the laws obeyed. Instead of the comparatively 
easy task of finding what effects will follow from a given 
law, the effects are now given and tlie law is required. 
We have to interpret the will by which the conditions 
of creation were laid down. 

Induction an Inverse Operation 

I have already asserted that induction is the inverse 
operation of deduction, but the difference is one of such 
great importance that I must dwell upon it. There are 
many cases in which we can easily and infallibly do a 
certain thing but may have much trouble in undoing it. 
A person may walk into the most complicated labyrinth 
or the most extensive catacombs, and turn hither and thither 
at his will ; it is when he wishes to return that doubt and 
difficulty commence. In entering, any path served him ; 
in leaving, he must select certain definite paths, and in this 
selection he must either trust to memory of the way he 
entered or else make an exhaustive trial of all possible 
ways. The explorer entering a new country makes sure 
his line of return by barking the trees. 

The same difficulty arises in many scientific processes. 
Given any two numbers, we may by a simple and infallible 
process obtain their product ; but when a large number 




is given it is quite another matter to determine its factors. 
Can the reader say what two numbers multiplied together 
will produce the number 8,616460,799? I think it 
unlikely that anyone but myself will ever know; for 
they are two large prime numbers, and can only be re- 
discovered by trying in succession a long series of prime 
divisors until the right one be fallen upon. The work 
would probably occupy a good computer for many weeks, 
but it did not occupy me many minutes to multiply the 
two factors together. Similarly there is no direct process 
for discovering whether any number is a prime or not ; it 
is only by exhaustively trying all inferior numbers which 
could be divisors, that we can show there is none, and the 
labour of the process would be intolerable were it not per- 
formed systematically once for all in the process known as 
the Sieve of Eratosthenes, the results being registered in 
tables of prime numbers. 

The immense difficulties which are encountered in the 
solution of algebraic equations afford another illustration. 
Given any algebraic factors, we can easily and infallibly 
arrive at the product ; but given a product it is a matter 
of infinite difficulty to resolve it into factors. Given any 
series of quantities however numerous, there is very little 
trouble in making an equation which shall have those 
quantities as roots. Let a, 6, c, d, &c., be the quantities ; 

then (x — a) (x — h) (x — c) (x - d) = o 

is the equation required, and we only need to multiply out 
the expression on the left hand by ordinary mles. But 
having given a complex algebraic expression equated to 
zero, it is a matter of exceeding difficulty to discover all 
the roots. Mathematicians have exhausted their highest 
powers in carrying the complete solution up to the fourth 
degree. In every other mathematical operation the inverse 
process is far more difficult than the direct process, sub- 
traction than addition, division than multiplication, evo- 
lution than involution ; but the difficulty increases vastly 
as the process becomes more complex. Differentiation, 
the direct process, is always capable of performance by 
fixed rules, but as tliese rules produce considerable variety 
of results, the inverse process of integration presents im- 
mense difficulties, and in an infinite majority of cases 
surpasses the oresent resources of mathematicians. There 


I i 

I * 




are no infallible and general rules for its accomplishment • 
It must be done by trial, by guesswork, or by remembering 
the results of differentiation, and using them as a guide 

Coming more nearly to our own immediate subject 
exactly the same difficulty exists in determining the law 
which certain things obey. Given a general mathematical 
expression, we can infallibly ascertain its value for anv 
required value of the variable. But I am not aware that 
mathematicians have ever attempted to lay down the rules 
of a process by which, having given certain numbers, one 
might discover a rational or precise formula from which 
they proceed. The reader may test his power of detectin^r 
a law, by contemplation of its results, if he, not bein^ a 
mathematicmn, will attempt to point out the law obeyed 
by the following numbers : 

30' 4a' 






510 ' 



These numbers are sometimes in low terms but un 
expectedly spring up to high terms; in absolute magnitude 
they are very variable. They seem to set all regularity 
and method at defiance, and it is hardly to be supposed 
that anyone could, from contemplation of the numbers 
have detected the relations between them. Yet they are 
derived from the most regular and symmetrical laws of 
relation, and are of the highest importance in mathematical 
analysis, being known as the numbers of Bernoulli 

Compare again the difficulty of decyphering with that 
ot cypliering. Anyone can invent a secret language, and 
with a little steady labour can translate the longest letter 
into the character. But to decypher the letter, having no 
key to the signs adopted, is a wholly different matter. 
As the possible modes of secret writing are infinite in 
number and exceedingly various in kind, there is no direct 
mode of discovery whatever. Repeated trial, guided more 
or less by knowledge of the customary form of cypher and 
resting entirely on the principles of probability and logical 
induction, is the only resource. A i)eculiar tact or skUl is 
requisite for the process, and a few men, such as Wallis or 
Wheatstone, have attained great success. 

Induction is the decyphering of the hidden meanincr of 
natural phenomena. Given events which happen in certain 





definite combinations, we are required to point out the 
laws which govern those combinations. Any laws being 
supposed, we can, with ease and certainty, decide whether 
the phenomena obey those laws. But the laws which may 
exist are infinite in variety, so that the chances are im- 
mensely against mere random guessing. The difficulty is 
much increased by the fact that several laws will usually 
be in operation at the same time, the effects of which 
are complicated together. The only modes of discovery 
consist either in exhaustively trying a great number of 
supposed laws, a process which is exhaustive in more 
senses than one, or else in carefully contemplating the 
effects, endeavouring to remember cases in which like 
effects followed from known laws. In whatever manner 
we accomplish the discovery, it must be done by the more 
or less conscious application of the direct process of 

The Logical Alphabet illustrates induction as well as 
deduction. In considering the Indirect Process of Inference 
we found that from certain propositions we could infallibly 
determine the combinations of terms agreeing with those 
premises. The inductive problem is just the inverse. 
Having given certiuu combinations of terms, we need to 
ascertain the propositions with which the combinations are 
consistent, and from which they may have proceeded. 
Now, if the reader contemi)lates the following combina- 

ABC ohG 

aBC ahc, 

he will probably remember at once that they belong to the 
premises A = AB, B = BC (p. 92). If not, he will require 
a few trials before he meets with the right answer, and 
every trial will consist in assuming certain laws and 
observing whether the deduced results agree with the data. 
To test the facility with which he can solve this inductive 
problem, let him casually strike out any of the combina- 
tions of the fourth column of the Logical Alphabet, (p. 94), 
and say what laws the remaining combinations obey, 
observing that every one of the letter-terms and their 
negatives ought to appear in order to avoid self-contradic- 
tion in the premises (pp.74, in). Let him say, for 
instance, what laws are embodied in the combinations 






Abe ahQ. 

The difficulty becomes much greater when more terms 
enter into the combinations. It would require some little 
examination to ascertain the complete conditions fulfilled 
in the combinations 

AC<5 ahCe 

aBC« abcK 

The reader may discover easily enough that the principal 
laws are C = «, and A = A^; but he would hardly discover 
without some trouble the remaining law, namely, that 
BD = BD«. 

The difficulties encountered in the inductive investigations 
of nature, are of an exactly similar kind. We seldom 
observe any law m uninternipted and undisguised opera- 
tion. The acuteness of Aristotle and the ancient Greeks 
did not enable them to detect that all terrestrial bodies 
tend to fall towards the centre of the earth. A few nights 
of observation might have convinced an astronomer 
viewing the solar system from its centre, that the planets 
travelled round the sun ; but the fact that our place of 
observation is one of the travelling planets, so conS)licates 
the apparent motions of the other bodies, that it required 
all the sagacity of Copernicus to prove the real simplicity 
of the planetary, system. It is the same throughout 
nature; the laws may be simple, but their combined 
effects are not simple, and we have no clue to guide us 
through their intricacies. " It is the glory of God," said 
Solomon, " to .conceal a thing, but the glory of a king to 
search it out" The laws of nature are the invaluable 
secrets which God has hidden, and it is the kingly pre- 
rogative of the philosopher to search them out by industry 
and sagacity. ^ 

Inductive Problems for Solution hy the Reader. 

In the first edition (vol ii. p. 370) I gave a logical 
problem involving six terms, and requested renders to 
discover the laws governing the combinations given. I 
received satisfactory replies from readers both in the 
United States and in England. I formed the combina- 


▼ II.] 



tions deductively from four laws of correction, but my 
correspondents found that three simpler laws, equivalent 
to the four more complex ones, were the best answer ; these 
laws ai-e as follows : a = ac, h = cd, d = Ef. 

In case other readers should like to test their skill in the 
inductive or inverse problem, I give below several series 
of combinations forming problems of graduated difficulty. 

Pmo.vlbm I. 


a b C D < 

A he D 

a b C d B 

L B e 

fiBO D 

a b e l> e 

A b 

aBe D 

a b e d E 

a BC 

a B e d 
a b Cd 


Prublem IX, 


Pkoblkm Tt. 




ABc D e P 
A b C D e/ 
Abe D E/ 
A b c D e / 
A b ed E F 
A b c d e F 
aBe D E P 
Be D e P 
a Bed £ F 

A b C 
a B C 

A B fi 

A B Cd« 
A B e d e 
a B C d e 
abC D E 


6C D E P 

A b C 

(t ft C D e F 

a B C 


b C D «/ 
a b e D « / 

a b e 

A b e D e 
aBC d E 
• 6 C d B 

■ b e D E y 
abode ¥ 

Pkoilbm it 

Pboblbm X. 



A be D 


a B e d 


ABe D B/ 

« b C d 



ABC de 

AbC De F 

AB e d e 

A b c D e F 



aBC D E^ 

A bed B 

aB e D K/ 


a bC D e F 


aB D < 

a b C d e F 

A BCd 

a B d e 

a b e D e / 

A Bed 

aB e D< 


Induction of Simple Identities, 

Many important laws of nature are expressible in the 
form of simple identities, and I can at once adduce them 
as examples to illustrate what I have said of the difficulty 
of the inverse process of induction. Two phenomena are 
conjoined. Thus all gravitating matter is exactly co- 
incident with all matter possessing inertia; where one 







property appears, the other likewise appears. All crystals 
of the cubical system, are all the crystals which do not 
doubly refract light. All exogenous plants are, with some 
exceptions, those which have two cotyledons or seed-leaves. 

A little reflection will show that there is no direct and 
infallible process by which such complete coincidences 
may be discovered. Natural objects are aggregates of 
many qualities, and any one of those qualities may prove 
to be in close connection with some others. If each of a 
numerous group of objects is endowed with a hundroc 
distinct physical or chemical qualities, there will be no 
less tlian I (lOO X 99) or 4950 pairs of qualities, which 
may be connected, and it will evidently be a matter of 
great intricacy and labour to ascertain exactly which 
qualities are connected by any simple law. 

One principal source of difficulty is that the finite powei-s 
of the human mind are not sufficient to compare by a 
single act any large group of objects with another large 
group. We cannot hold in the conscious possession of the 
mind at any one moment more than five or six different 
ideas. Hence we must tieat any more complex group by 
successive acts of attention. Tlie reader will perceive by 
an almost individual act of comparison that the words 
Roma and Mora contain the same letters. He may 
perhaps see at a glance whether the same is true of 
Causal and Casual^ and of Logica and Caligo. To assure 
himself that the letters in Astronomers make No more 
stars, that Serpens in akuleo is an anagram of Joannes 
Keplenis, or Great gun do us a sum an anagram of Au- 
gustus de Morgan, it will cei*tainly be necessary to break 
up the act of comparison into several successive acts. The 
process will acquire a double character, and will consist in 
ascertaining that each letter of the first group is among 
the letters of the second group, and vice versd, that each 
letter of the second is among those of the first group. 
In the same way we can only prove that two long lists of 
names are identical, by showing that each name in one 
list occurs in the other, and vice versd. 

This process of comparison really consists in establishing 
two partial identities, which are, as already shown (p. 58), 
equivalent in conjunction to one simple identity. We 
first ascertain the truth of the two propositions A = AB, 




B = AB, and we then rise by substitution to the single 
law A = B. ^ 

There is another process, it is true, by which we may 
get to exactly the same result ; for the two propositions 
A = AB, o = oi are also equivalent to the simple identity 
A = B. If then we can show that all objects included 
under A are included under B, and also that all objects 
not included under A are not included under B, our pur- 
pose is effected. By this process we should usually com- 
pare two lists if we are allowed to mark them. For each 
name in the first list we should strike off one in the second, 
and if, when the first list is exhausted, the second list is 
also exhausted, it follows that all names absent from the 
first must be absent from the second, and the coincidence 
must be complete. 

These two modes of proving an identity are so closely 
allied that it is doubtful how far we can detect any differ- 
ence in their powers and instances of application. The 
first method is perhaps more convenient when the pheno- 
mena to be compared are rare. Thus we prove that all 
the musical concords coincide with all the more simple 
numerical ratios, by showing that each concord arises from 
a simple ratio of undulations, and then showing that eack 
simple ratio gives rise to one of the concords. To examine 
all the possible cases of discord or complex ratio of 
undulation would be impossible. By a happy stroke of 
induction Sir John Herschel discovered that all crystals 
of quartz which cause the plane of polarization of light 
to rotate are precisely those crystals which have plagi- 
hedral faces, that is, oblique faces on the comers of the 
prism unsymmetrical with the ordinary faces. This 
singular relation would be proved by observing that all 
plagihedral crystals possessed the power of rotation, and 
Tfice versd all crystals possessing this power were plagi- 
hedral But it might at the same time be noticed that 
all ordinary crystals were devoid of the power. There is 
no reason why we should not detect any of the four pro- 
positions A = AB, B = AB, a = ah, b = ah, all of which- 
follow from A =» B (p. 115). 

Sometimes the terms of the identity may be singular 
objects ; thus we observe that diamond is a combustible gem, 
and being unable to discover any other that is, we affirm — 



[on AW 

Diamond = combustible gem. 
In a similar manner we ascertain that 

Mercury = metal liquid at ordinary temperatures, 
Substance of least density = substance of least atomic 
Two or three objects may occasionally enter into the 
induction, as when we learn that 

Sodium -I* potassium = metal of less density than 

Venus •!• Mercury •!• Mars = major planet devoid of 

' f 

IndvAiion. of Partial Identities, 

We found in the last section that the complete identity 
of two classes is almost always discovered not by direct 
observation of the fact, but by first establishing two 
partial identities. There are also a multitude of cases in 
which the partial identity of one class with another is the 
only relation to be discovered. Thus the most common of 
all inductive inferences consists in establishing the fact 
that all objects having the properties of A have also those 
of B, or that A = AB. To ascertain the truth of a pro- 
position of this kind it is merely necessary to assemble 
together, mentally or physically, all the objects included 
under A, and then observe whether B is present in each 
of them, or, which is the same, whether it would be im- 
possible to select from among them any not-B. Thus, if 
we mentally assemble together all the heavenly bodies 
which move with apparent rapidity, that is to say, the 
planets, we find that they all possess the property of not 
scintillating. We cannot analyse any vegetable substance 
without discovering that it contains carbon and hydrogen, 
but it is not true that all substances containing carbon 
and hydrogen are vegetable substances. 

The great mass of scientific truths consists of propo- 
sitions of this form A = AB. Thus in astronomy we learn 
that all the planets are spheroidal bodies ; that they all 
revolve in one direction round the sun ; that they all shine 
by reflected light; that they all obey the law of gravi- 
tation. But of course it is not to be asserted that all 
bodies obeying the law of gravitation, or shining by 




reflected light, or revolving in a particular direction, or 
being spheroidal in form, are planets. In other sciences 
we have immense numbers of propositions of the same 
form, as, for instance, all substances in becoming gaseous 
absorb heat ; all metals are elements ; they are' all good 
conductors of heat and electricity ; all the alkaline metals 
are monad elements; all foraminifera are marine organ- 
isms ; all parasitic animals are non-mammalian ; lightning 
never issues from stratous clouds; pumice never occurs 
where only Labrador felspar is present ; milkmaids do 
not suffer from small-pox ; and, in the works of Darwin, 
scientific importance may attach even to such an appa- 
rently trifling observation as that " white tom-cats having 
blue eyes are deaf." 

The process of inference by which all such truths are 
obtained may readily be exhibited in a precise symbolic 
form. We must have one premise specifying in a dis- 
junctive form all the possible individuals which belong 
to a class ; we resolve the class, in short, into its con- 
stituents. We then need a number of propositions, each 
of which affirms that one of the individuals possesses a 
certain property. Thus the premises must be of the 


A = B f. C 

i D .[. 
C = CX 

+ P + Q 


Q = QX. 

Now, if we substitute for eacli alternative of the first 

premise its description aa found among the succeedin*^ 

premises, we obtain " 

A = BX + CX + -H PX .|. QX 


A = (B .|. C + .|. Q)X 

But for the aggregate of alternatives we may now 
substitute their equivalent as given in the firat premise, 
namely A, so that we get the required result : 

A = AX. 
We should have reached the same result if the first 
premise had been of the form 

A = AB .|. AC ^' + AQ. 

K 2 

\> I 






We can always prove a proposition, if we find it more 
convenient, by proving its equivalent, fo assert that all 
not-B*s are not-A's, is exactly the same as to assert that all 
A's are B's. Accordingly we may ascertain that A - AB by 
first ascertaining that b «- ab. If we observe, for instance, 
that all substances which are not solids are also not capable 
of double refraction, it follows necessarily that all double 
refracting substances are solids. We may convince our- 
selves that all electric substances are nonconductors of 
electricity, by reflecting that all good^ conductors do not, 
and in fact cannot, retain electric excitation. When we 
come to questions of probability it will be found desirable 
to prove, as far as possible, both the original proposition 
and its equivalent, as there is then an increased area of 

The number of alternatives which may arise in the 
division of a class varies greatly, and may be any number 
from two upwards. Thus it is probable that every sub- 
stance is either magnetic or diamagnetic, and no substance 
can be both at the same time. The division then must be 
made in the form 

A = ABc + AbG. 

If now we can prove that all magnetic substances are 
capable of polarity, say B = BD, and also that all dia- 
magnetic substances are capable of polarity, C = CD, it 
follows by substitution that all substances are capable of 
polarity, or A = AD. We commonly divide the class sub- 
stance into the three subclasses, solid, liquid, and gas ; and 
if we can show that in each of these forms it obeys Carnot's 
thermodynamic law, it follows that all substances obey 
that law. Similarly we may show that all vertebrate 
animals possess red blood, if we can show separately that 
fish, reptiles, birds, marsupials, and mammals possess red 
blood, there being, as far as is known, only five principal 
subclasses of vertebrata. 

Our inductions will often be embarrassed by exceptions, 
real or apparent We might affirm that all gems are in- 
combustible were not diamonds undoubtedly combustible. 
Nothing seems more evident than that all the metals are 
opaque until we examine them in fine films, when gold and 
silver are found to be ti-ausparent. All plants absorb 
carbonic acid except certain fungi ; all the bodies of the 




planetary system, have a progressive motion from west to 
east, except the satellites of Uranus and Neptune. Even 
some of the profoundest laws of matter are not quite 
universal ; all solids expand by heat except india-nibber, 
and possibly a few other substances ; all liquids which have 
been tested expand by heat except water below 4° C. and 
fused bismuth; all gases have a coefficient of expansion 
increasing with the temperature, except hydrogen. In 
a later chapter I shall consider how such anomalous 
cases may be regarded and classified ; here we have only to 
expi-ess them in a consistent manner by our notation. 
Let us take the case of the transparency of metals, and 

D = iron 

E, F, &c. = copper, lead, &c. 

X = opaque. 

C D |. E, &c. 

Now evidently 


assign the terms thus : — 

A s= meoal 
B = gold 
C = silver 
Our premises will be 
A = B 
B = Ba; 
C = Cte 
and so on for the rest of the metals. 

Abe = (D ^. E .|. F + 
and by substitution as before we shall obtain 

Abe = AbcX, 

or in words, "All metals not gold nor silver are opaque •/ 
at the same time we have 

A(B + C) = AB .|. AC = ABa; -I- AGx = A(B |. C)a;. 
or " Metals which are either gold or silver are not opaque." 
In some cases the problem of induction assumes a much 
higher degree of complexity. If we examine the properties 
of crystallized substances we may find some properties 
which are common to all, as cleavage or fracture in definite 
planes ; but it would soon become requisite to break up 
the class into several minor ones. We should divide 
crystals according to the seven accepted systems — and we 
should then find that crystals of each system possess 
many common properties. Thus crystals of the Regular 
or Cubical system expand equally by heat, conduct heat 
and electricity with uniform rapidity, and are of like 
elasticity in all directions; they have but one index of 








■ > 

refraction for light ; aud every facet i^ repeated in like 
relation to each of the three axes. Crystals of the system 
having one principal axis will be found to possess the 
various physical powers of conduction, refraction, elas- 
ticity, &c., uniformly in directions perpendicular to the 
principal axis ; in other directions their properties vary 
according to complicated laws. The remaining systems 
in which the crystals possess three unequal axes, or have 
inclined axes, exhibit still more complicated results, the 
effects of the crystal upon light, heat, electricity, &c., 
varying in all directions. But when we pursue induction 
into the intricacies of its application to nature we really 
enter upon the subject of classification, which we must 
take up again in a later part of this work. 

Solution of the Inverse or Inductive Problem, involving 

Tioo Classes. 

It is now plain that Induction consists in passing back 
from a series of combinations to the laws by which such 
combinations are governed. The natural law that all 
metals are conductors of electricity really means that in 
nature we find three classes of objects, namely — 

1. Metals, conductors ; 

2. Not-metals, conductors ; 

3. Not-metals, not-conductors. 

It comes to the same thing if we say that it excludes the 
existence of the class, "metals not-conductors." In the 
same way every other law or group of laws will really 
mean the exclusion from existence of certain combinations 
of the things, circumstances or phenomena governed by 
. those laws. Now in logic, strictly speaking, we treat not 
the phenomena, nor the laws, but the general forms of the 
laws ; and a little consideration will show that for a finite 
number of things the possible number of forms or kinds 
of law governing them must also be finite. Using general 
terms, we know that A and B can be present or absent in 
four ways and no more — thus : 

AB, Ab, oB, ab; 
therefore every possible law which can exist concerning 
the relation of A and B must be marked by the exclusion 
•f one or more of the <».bove combinations. The number 




of possible laws then cannot exceed the number of selec- 
tions which we can make from these four combinations. 
Since each combination may be present or absent, the 
number of cases to be considered is 2 x 2 x 2 x 2, or sixteen ; 
and these cases are all shown in the following table, in 
which the sign o indicates absence or non-existence of the 
combination shown at tHe left-hand column in the same 
line, and the mark i its presence : — 

























































' 'Thus in colunm sixteen we find that all the conceivable 
combinations are present, which means that there are no 
special laws in existence in such a case, and that the 
combinations are governed only by the universal Laws of 
Identity and Difference. The example of metals and 
conductors of electricity would be represented by the 
twelfth column ; and every other mode in which two 
things or qualities might pi'esent themselves is shown in 
one or other of the columns. More than half the cases 
may indeed be at once rejected, because they involve the 
entire absence of a term or its negative. It has been 
shown to be a logical principle that every term must have 
its negative (p. 11 1), and when this is not the case, incon- 
sistency between the conditions of combination must exist. 
Thus if we laid down the two following propositions, 
" Graphite conducts electricity," and " Graphite does not 
conduct electricity," it would amount to asserting the 
impossibility of graphite existing at all ; or in general 
terms, A is B and A is not B result in destroying alto- 
gether the combinations containing A, a case shown in the 
fourth column of the above table. We therefore restrict 
our attention to those cases which may be represented in 
natural phenomena when at least two combinations are 
present, and which correspond to those columns of the 








table in which each of A, a, B, 6 appears. These cases 
are shown in the columns marked with an asterisk. 

We find that seven cases remain for examination, thus 
characterised — 

Four cases exhibiting three combinations, 
Two cases exhibiting two combinations. 
One case exhibiting four combinations.' 
It lias already been pointed out that a proposition of the 
form A = AB destroys one combination, AJ, so that this is 
the form of law applying to the twelfth column. But by 
changmg one or more of the terms in A = AB into its 
negative or by interchauging A and B, a and b, we obtain 
no less than eight different varieties of the one form ; thus— 

lathcase. 8th CMa. 15th ewe, t4thea8e. 

A = AB A = Aft a = aB a = ab 
b = ab B = aB b = Ab B = AB 

The reader of the preceding sections will see that each 
proposition in the lower line is logically equivalent to and 
18 m fact the contrapositive of, that above it (p. ^i) Thus 
the propositions A - A6 and B = aB both give the same 
combinations, shown in the eighth column of the table 
and trial shows that the twelfth, eighth, fifteenth and 
fourteenth columns are thus accounted for. We come to 
this conclusion then— The general form of proposition 
A --AB admits of four logically distinct varieties, eack 
capable of expression in two modes. 

In two columns of the table, namely the seventh and 

tenth, we observe that two combinations are missing 

Now a simple identity A = B renders impossible both Ab 

and aB, accounting for the tenth case ; and if we change 

B into b the identity A = J accounts for the seventh case 

Ihere may indeed be two other varieties of the simple 

identity, namely a = & and a = B ; but it has already 

been shown repeatedly that these are equivalent respec 

tively to A = B and A = 6 (p. 115). As the sixteenth 

column has already been accounted for as governed 

by no special conditions, we come to the following general 

conclusion :— The laws governing the combinations of two 

terms must be capable of expression either in a partial 

Identity or a simple identity ; the partial identity is capable 

of only four logically distinct varieties, and the simple 

^entity of two. Every logical relation between two terns 





must be expressed in one of these six forms of law, or 
must be logically equivalent to one of them. 

In short, we may conclude that in treating of partial 
and complete identity, we have exhaustively treated the 
modes in which two terms or classes of objects can be 
related. Of any two classes it can be said that one must 
either be included in the other, or must be identical with 
it, or a like relation must exist between one class and the 
negative of the other. We have thus completely solved 
the inverse logical problem concerning two terms.^ 

The Inverse Ijogical Problem involving Three Classes. 

No sooner do we introduce into the problem a third term 
C, than the investigation assumes a far more complex 
character, so that some readers may prefer to pass over 
this section. Three terms and their negatives may be 
combined, as we have frequently seen, in eight different 
combinations, and the effect of laws or logical conditions 
is to destroy any one or more of these combinations. Now 
we may make selections from eight things in 2" or 256 
ways; so that we have no less than 256 different cases to 
treat, and the complete solution is at least fifty times as 
troublesome as with two terms. Many series of com- 
binations, indeed, are contradictory, as in the simpler 
problem, and may be passed over, the test of consistency 
being that each of the letters A, B, C, a, 6, c, shall appear 
somewhere in the series of combinations. 

My mode of solving the problem was as follows: — 
Having written out the whole of the 256 series of com- 
binations, I examined them separately and struck out such 
as did not fulfil the test of consistency. I then chose 
some form of proposition involving two or three terms, 
and varied it in every possible manner, both by the 
circular interchange of letters (A, B, C into B, C, A and 
then into C, A, B), and by the substitution for any one or 
more of the terms of the corresponding negative terms. 

_ • 

* The contents of this and the following section nearly correspond 
with those of a paper read before the Manchester Literary and 
Philosophical Society on December 26th, 187 1. See Proceedings of 
the Society, vol. xi. pp. 65—68, and Memoirs, Third Series, voL r. 
pp. 119-130. 











For instance, the proposition AB = ABC can be first 
varied by circular interchange so as to give BC = BCA and 
then CA = CAB. Each of these three can then be thrown 
into eight varieties by negative change. Thus AB = ABC 
gives aB = aBC, Ab = A6C, AB = ABc, ab = ahC, and 
so on. Thus there may possibly exist no less than twenty- 
four varieties of the law having the general form 
AB = ABC, meaning th»nt whatever has the properties of 
A and B has those also of C. It by no means follows 
that some of the varieties may not be equivalent to others ; 
and trial shows, in fact, that AB = ABC is exactly the 
same in meaning as Ac = Abe or Be = Bca. Thus the law 
in question has but eight varieties of distinct logical mean- 
ing. I now ascertain by actual deductive reasoning which 
of the 256 series of combinations result from each of these 
distinct laws, and mark them off as soon as found. I then 
proceed to some other form of law, for instance A = ABC, 
meaning, that whatever has the qualities of A has those 
also of B and C. I find that it admits of twenty-four 
variations, all of which are found to be logically distinct ; 
the combinations being worked out, I am able to mark off 
twenty-four more of the list of 256 series. I proceed in 
this way to work out the results of every form of law 
which I can find or invent. If in the course of this work 
I obtain any series of combinations which had been pre- 
viously marked off, I learn at once that the law giving 
these combinations is logically equivalent to some law 
previously treated. It may be safely inferred that every 
variety of the apparently new law will coincide in meaning 
with some variety of the former expression of the same 
law. I have sufficiently verified this assumption in some 
cases, and have never found it lead to error. Thus as 
AB = ABC is equivalent to Ac = Abe, so we find that 
ab = ahC is equivalent to ac = ocB. 

Among the laws treated were the two A = AB and 
A — B which involve only two terms, because it may of 
course happen that among three things two only are in 
spedial logical relation, and the third independent; and 
the series of combinations representing such cases of re- 
lation are sure to occur in the complete enumeration. All 
single propositions which I could invent having been 
treated, pairs of propositions were next investigated. Thus 

wo have the relations, " All A's are B's and all B's are 
C's," of which the old logical syllogism is the development. 
We may also have " all A's are all B's, and all B*s are C's," 
or even "all A's are all B's, and all B's are all C's." All 
such premises admit of variations, greater or less in 
number, the logical distinctness of which can only be 
determined by trial in detail. Disjunctive propositions 
either singly or in pairs were also treated, but were often 
found to be equivalent to other propositions of a simpler 
form ; thus A = ABC -I- Abe is exactly the same in meaning 
as AB = AC. 

This mode of exhaustive trial bears some analogy to 
that ancient mathematical process called the Sieve of 
Eratosthenes. Having taken a long series of the natural 
numbers, Eratosthenes is said to have calculated out in 
succession all the multiples of every number, and to 
have marked them off, so that at last the prime numbers 
alone remained, and the factoi*s of every number were 
exhaustively discovered. My problem of 256 series of 
combinations is the logical analogue, the chief points of 
difference being that there is a limit to the number of cases, 
and that prime numbers have no analogue in logic, since 
every series of combinations corresponds to a law or group 
of conditions. But the analogy is perfect in the point that 
they are both inverse processes. There is no mode of 
ascertaining that a number is prime but by showing that 
it is not the product of any assignable factors. So there 
is no mode of ascertaining what laws are embodied in any 
series of combinations but trying exhaustively the laws 
which would give thenL Just as the results of Erato- 
sthenes' method have been worked out to a great extent 
and registered in tables for the convenience of other 
mathematicians, I have endeavoured to work out the 
inverse logical problem to the utmost extent which is at 
present practicable or ujefuL 

I have thus found that there are altogether fifteen con- 
ditions or series of conditions which may govern the com- 
binations of three terms, forming the premises of fifteen 
essentially different kinds of arguments. The following 
table contains a statement of these conditions, together 
with the numbers of combinations which are contradicted 
or destroyed by each, and the numbers of logically distinct 




variations of which the law is capable. There might be 
also added, as a sixteenth case, that case where no special 
logical condition exists, so that aU the eight combinations 









riuposltione expressing tb« genenl 
type of ihe logical conditiona. 

1 Nnmberofdi*. 
1 tinct logical 

f Namber of 
by each. 


A = B 



A = AB 


A = B, B = C 



A = B. B = BC 



A = AB, B a BC 






A = ABC 






A = AB. aB = aBe 


A = ABC, ab m abC 



AB=sABC. ab=:abe 


AB = AC 



A =: BC 1- Afte 

A = BC j. be 



A -ABC. a=sBc.|. 6(7 


There are sixty-three series of combinations derived from 
self-contradictory premises, which with 192, the snm of 
the numbers of distinct logical variations stated in the 
third column of the table, and with the one case where 
there are no conditions or laws at all, make up the whole 
conceivable number of 256 series. 

We learn from this table, for instance, that two pro- 
positions of the form A = AB, B = BC, which are such 
as constitute the premises of the old syllogism Barbara 
exclude as impossible four of the eight combinations in 
which three terms may be united, and that these proposi- 
tions are capable of taking twenty-four variations by tmns- 
positions of the terms or the introduction of natives 
This table then presents the results of a complete analysis 
of all the possible logical relations arising in the case of 
three terms, and the old syllogism forms but one out of 
fifteen typical forms. GeneraUy speaking, every form can 
h! T^^l^.^ '""^ apparently different propositions ; thus 
the fourth type A = B, B = BC may appeaV in the form 
A - Ai50, a = «J, or again m the form of three proposi- 
tions A = AB, B =BC, aB = aBc; but all these seHf 
premises yield identically the same series of combinations, 




and are therefore of equivalent logical meaning. The fifth 
type, or Barbara, can also be thrown into the equivalent 
forms A « ABC, aB = aBC and A = AC, B = A I- aBC. 
In other cases I have obtained the very same logical 
conditions in four modes of statements. As regards mere 
appearance and form of statement, the number of possible 
premises would be very great, and difficult to exhibit 

The most remarkable of all the types of logical condition 
is the fourteenth, namely, A = BC I- be. It is that which 
expresses the division of a genus into two doubly marked 
species, and might be illustrated by the example—" Com- 
ponent of the physical universe = matter, gravitating, or 
not-matter (ether), not-gravitating." It is capable of only 
two distinct logical variations, namely, A = BC •!• he and 
A = Be + 6C. By transposition or negative change of the 
letters we can indeed obtain six different expressions of 
each of these propositions ; but when their meanings are 
analysed, by working out the combinations, they are found 
to be logically equivalent to one or other of the above two. 
Thus the proposition A = BC •!• he can be written in any 
of the following five other modes, 

o = iC I- Be. B = CA .|. ca, J = cA I- Ca, 
C = AB .|. ah, c = aB + A*. 

I do not think it needful to publish at present the com- 
plete table of 193 series of combinations and the premises 
corresponding to each. Such a table enables us by mere 
inspection to learn the laws obeyed by any set of com- 
binations of three things, and is to logic what a table of 
factors and prime numbers is to the theory of numbers, or 
a table of integrals to the higher mathematics. The table 
already given (p. 140) would enable a person with but little 
labour to discover the law of any combinations. If there 
be seven combinations (one contradicted) the law must be 
of the eighth type, and the proper variety will be apparent. 
If there be six combinations (two contradicted), either the 
second, eleventh, or twelfth type applies, and a certain 
number of trials will disclose the proper type and variety. 
If there be but two combinations the law must be of the 
third type, and so on. 

The above investigations are complete as regards the 
possible logical relations of two or three terms. But 


M If 

I / 









when we attempt to apply the same kind of method to 
the relations of four or more terms, the labour becomes 
impracticably gi-eat Four terms give sixteen combinations 
compatible with the laws of thought, and the number of 
possible selections of combinations is no less than 2^* or 
65,536. The following table shows the extraordinary 
manner in which the number of possible logical relations 
increases with the number of terms involved. 

Namber of 





Namber of 
possible com< 





Nnmber of pouible selections of combinations 

corrasponding to consistent or inconsistent 

I(^cal relations. 



. , 4,a94,967,a96 


Some years of continuous labour would be required to 
ascertain the types of laws which may govern the com- 
binations of only four things, and but a small part of such 
laws would be exemplified or capable of practical appli- 
cation in science. The purely logical inverse problem, 
whereby we pass from combinations to their laws, is 
solved in the preceding pages, as far as it is likely to be 
for a long time to come ; and it is almost impossible that 
it should ever be carried more than a single step 
further. ^ ^ 

In the first edition, vol I p. 158, I stated' that I had not 
been able to discover any mode of calculating the number 
of cases in which inconsistency would be implied in the 
selection of combinations from the Logical Alphabet. The 
logical complexity of the problem appeared to be so great 
that the ordinary modes of calculating numbers of com- 
binations failed, in my opinion, to give any aid, and 
exhaustive examination of the combinations in detail 
seemed to be the only method applicable. This opinion, 
however, was mistaken, for both Mr. R. B. Hayward, of 
Harrow, and Mr. W. H. Brewer have calculated the 
numbers of inconsistent cases both for three and for four 
terms, without much difficulty. In the case of four 
terms they find that there are 1761 inconsistent selections 
and 63,774 consistent, which with one case wliere no 






condition exists, make up the total of 65,536 possible 

The inconsistent cases are distributed in the manner 
shown in the following table ; — 

Number of 

Combi nations 



• 3 4 5 6 





Namber of 



I x6 

iia 35« 536 448 9*4 




0, &e. 

When more than eight combinations of the Logical 
Alphabet (p. 94, column V.) remain unexcluded, there cannot 
be inconsistency. The whole numbers of ways of selecting 
o, 1,2, &c., combinations out of 16 are given in the 17th 
line of the Arithmetical Triangle given further on in the 
Chapter on Combinations and Permutations, the sum of 
the numbers in that line being 65,536. 

Professor Clifford on the Types of Compound Statement 

involving Four Classes. . 

In'the first edition (vol i. p. 163), I asserted that some 
years of labour would be required to ascertain even the 
precise number of types of law governing the combinations 
of four classes of things. Though I still believe that some 
years' labour would be required to work out the types 
themselves, it is clearly a mistake to suppose that the 
numbers of such types cannot be calculated with a reason- 
able amount of labour. Professor W. K Clifford having 
actually accomplished the task. His solution of the 
numerical problem involves the use of a complete new 
system of nomenclature and is far too intricate to be fully 
described here. I can only give a brief abstract of the 
results, and refer readers, who wish to follow out the 
reasoning, to the Proceedings of the Literary and Philo- 
sophical Society of Manchester, for the 9th January, 1877, 
voL xvi., p. 88, where Professor Clifford's paper is printed 
in full. 

By a simple statement Professor Clifford means the denial 
of the existence of any single combination or crossr 






I i 

IN* • 

iill " 

division, of the classes, as in ABCD = o, or AbCd — a 
The denial of two or more such combinations is called a 
compound statement, and is further said to be twofold, 
threefold, &c., according to the number denied. Thus 
ABC = o is a twofold compound statement in regard to 
four classes, because it involves both ABCD = o and 
ABC<i = o. When two compound statements can be 
converted into one another by interchange of the classes, 
A, B, C, D, with each other or with their complementary 
classes, a, h, c, d, they are called similar, and all similar 
statements are said to belong to the same type. 

Two statements ai'e called complementary when they 
deny between them all the sixteen combinations without 
both denying any one ; or, which is the same thing, when 
each denies just those combinations which the other 
permits to exist It is obvious that when two statements 
are similar, the complementary statements will also be 
similar, and consequently for every type of n-fold statement, 
there is a complementary type of (i6 — 7t)-fold statement. 
It follows that we need only enumerate the types as far as 
the eighth order; for the types of more-than -eight-fold 
statement will already have been given as complementary 
to types of lower orders. 

One combination, ABCD, may be converted into another 
AhCd by interchanging one or more of the classes with 
the complementary classes. The number of such changes 
is called the distance, which in the above case is 2. In 
two similar compound statements the distances of the 
combinations denied must be the same ; but it does not 
follow that when all the distances are the same, the state- 
ments are similar. There is, however, ouly one example 
of two dissimilar statements having the same distances. 
When the distance is 4, the two combinations are said to 
be obverse to one another, and the statements denying them 
are called obverse statements, as in ABCD = o and ahcd = o 
or again AbCd = o and aBcD = o. When any one com- 
bination is given, called the origin, all the others may be 
grouped in respect of their relations to it as foUows : — Four 
are at distance one from it, and may be called proocimaies ; 
six are at distance two, and may be called mediates ; four 
are at distance three, and may be called ultimMes ; finally 
the obverse is at distance /our. 




Origin and 
four proximatea. 








Obverse and 
four ultlmatea. 

abcD—abcd — aBcd 




It will be seen that the four proxiraates are respectively 
obverse to the four ultimates, and that the mediates form 
three pairs of obverses. Every proximate or ultimate is 
distant I and 3 respectively from such a pair of mediates. 

Aided by this system of nomenclature Professor Cliflford 
proceeds to an exhaustive enumeration of types, in which 
It IS impossible to follow him. The results are as follows — 

I -fold statements 





/ »> »f 

8-fold statements 











I type 
4 types 










Now as each seven-fold or less-than-seven-fold statement 
IS complementary to a nine-fold or more-than- nine-fold 
statement, it follows that the complete number of types 
will be 159 X 2 + 78 = 396. 

It appears then that the types of statement concernincr 
four classes are only about 26 times as numerous as those 
concerning three classes, fifteen in number, although the 
number of possible combinations is 256 times as great. 

Professor Clifford informs me that the knowledge of the 
possible groupings of subdivisions of classes which he 
obtained by this inquiry has been of service to him in 
some applications of hyper-elliptic functions to which he 
lias subsequently been led. Professor Cayley has since 
expressed his opinion that this line of investigation should 
be followed out, owing to the bearing of the theory of 
compound combinations upon the higher geometry.^ It 
seems likely that many unexpected points of connection 

fitK ^*'^«^*«^«y '^ Manchester Literary and Philosophical Soci^m, 
oth Febniaiy, 1877, vol. xvl, p. 1 13. 









will in time be disclosed between the sciences of logic 
and mathematics. 

Distinction between Perfect and Imperfect IndiLction. 

We cannot proceed with advantage befoi-e noticing the 
extreme difference which exists between cases of perfect 
and those of imperfect induction. We call an induction 
perfect when all the objects or events which can possibly 
come under the class treated have been examined. But 
in the majority of cases it is impossible to collect together, 
or in any way to investigate, the properties of all portions 
of a substance or of all the individuals of a race. Tlie 
number of objects would often be practically infinite, and 
the greater part of them might be beyond our reach, in 
the interior of the earth, or in the most distant parts of 
the Universe. In all such cases induction is imperfeety 
and is affected by more or less uncertainty. As some 
writers have fallen into much error concerning the func- 
tions and relative importance of these two branches of 
reasoning, I shall have to point out that — 

1. Perfect Induction is a process absolutely requisite, 

l)oth in the performance of imperfect induction and 
in the treatment of large bodies of facts of wliich 
our knowledge is complete. 

2. Imperfect Induction is founded on Perfect Induction, 

but involves another process of inference of a 
widely different character. 
It is certain that if I can draw any inference at all 
concerning objects not examined, it must be done on the 
data aflbrded by the objects which have been examined. 
If I judge that a distant star obeys the law of gravity, 
it must be because all other material objects sufficiently 
known to me obey that law. If I venture to assert that 
all ruminant animals have cloven hoofs, it is because all 
ruminant animals which have come under my notice have 
cloven hoofs. On the other hand, I cannot safely say 
that all cryptogamous plants possess a purely cellular 
structure, because some cryptogamous plants, which have 
been examined by botanists, have a partially vascular 
structure. The probability that a new cryptogam will be 
cellular only can be estimated, if at all, on the ground of 


the comparative numbers of known cryptogams which 
are and are not cellular. Thus the first step in every 
induction will consist in accurately summing up the 
number of instances of a particular phenomenon which 
have fallen under our observation. Adams and Leverrier, 
for instance, must have inferred that the undiscovered 
planet Neptune would obey Bode's law, because all the 
planets known at that time obeyed it. On what principles 
the passage from the known to the apparently unknown 
is warranted, must be carefully discussed in the next sec- 
tion, and in various parts of this work. 

It would be a great mistake, however, to suppose that 
Perfect Induction is in itself useless. Even when the 
enumeration of objects belonging to any class is complete, 
and admits of no inference to unexamined objects, the 
statement of our knowledge in a general proposition is a 
process of so much importance that we may consider it 
necessary. In many cases we may render our investiga- 
tions exhaustive ; all the teeth or bones of an animal ; all 
the cells in a minute vegetable organ ; all the caves in a 
mountain side ; all the strata in a geological section ; all 
the coins in a newly found hoard, may be so completely 
scrutinized that we may make some general assertion 
concerning them without fear of mistake. Every bone 
might be proved to cont^ain phosphate of lime ; every cell 
to enclose a nucleus ; every cave to hide remains of extinct 
animals ; every stratum to exhibit signs of marine origin ; 
every coin to be of Roman manufacture. These are cases 
where our investigation is limited to a definite portion of 
matter, or a definite area on the earth's surface. 

There is another class of cases where induction is 
naturally and necessarily limited to a definite number of 
alternatives. Of the regular solids we can say without the 
least doubt that no one has more than twenty faces, thirty 
edges, and twenty comers ; for by the principles of geometry 
we learn that there cannot exist more than five regular 
solids, of each of which we easily observe that the above 
statements are true. In the theory of numbers, an endless 
variety of perfect inductions might be made ; we can show 
that no number less than sixty possesses so many divisors, 
and the like is true of 360 ; for it does not require a great 
amount of labour to ascertain and count all the divisors 

L 2 






of numbers up to sixty or 360. I can assert that between 
60,041 and 60,077 no prime number occurs, because the 
exhaustive examination of those who have constructed 
tables of prime numbers proves it to be so. 

In matters of human appointment or history, we can 
frequently have a complete limitation of the number of 
instances to be included in an induction. We might show 
that the propositions of the third book of Euclid treat only 
of circles ; that no part of the works of Galen mentions the 
fourth figure of the syllogism ; that none of the other kings 
of England reigned so long as George III.; that Magna 
Charta has not been repealed by any subsequent statute ; 
that the price of corn in England has never been so high 
since 1847 as it was in that year; that the price of the 
English funds has never been lower than it was on the 
23rd of January, 1798, when it fell to 47 J. 

It has been urged against this process of Perfect Induc- 
tion that it gives no new information, and is merely a 
summing up in a brief form of a multitude of particulars. 
But mere abbreviation of mental labour is one of the most 
important aids we can enjoy in the acquisition of knowledge. 
The powers of the human mind ai*e so limited that multi- 
plicity of detail is alone sufficient to prevent its progress 
in many directions. Thought would be practically impos- 
sible if every separate fact had to be separately thought 
and treated. Economy of mental power may be considered 
one of the main conditions on which our elevated intellectual 
position depends. Mathematical processes are for the most 
part but abbreviations of the simpler acts of addition and 
subtraction. The invention of logarithms was one of the 
most striking additions ever made to human power : yet it 
was a mere abbreviation of operations which could have 
been done before had a sufficient amount of labour been 
available. Similar additions to our power will, it is hoped, 
be made from time to time ; for the number of mathematical 
problems hitherto solved is but an indefinitely small 
fraction of those which await solution, because the labour 
they have hitherto demanded renders them impracticable. 
So it is throughout all regions of thought. The amount 
of our knowledge depends upon our power of bringing it 
within practicable compass. Unless we arrange and 
classify facts and condense them into general truths, they 




soon surpass our powers of memory, and serve but to 
confuse. Hence Perfect Induction, even as a process of 
abbreviation, is absolutely essential to any high degree of 
mental achievement 

Transition frovi Perfect to Imperfect Induction. 

It is a question of profound difficulty on what grounds 
we are warranted in inferring the future from the present, 
or the nature of undiscovered objects from those which we 
liave examined with our senses. We pass from Perfect to 
Imperfect Induction when once we allow our conclusion to 
apply, at all events apparently, beyond the data on which 
it was founded. In making such a step we seem to gain a 
net addition to our knowledge ; for we learn the nature of 
what was unknown. We reap where we have never sown. 
We appear to possess the divine power of creating know- 
ledge, and reaching with our mental arms far beyond the 
sphere of our own observation. I shall have, indeed, U\ 
point out certain methods of reasoning in which we dt. 
pass altogether beyond the sphere of the senses, and 
acquire accurate knowledge which observation could 
never have given ; but it is not imperfect induction that 
accomplishes such a task. Of imperfect induction itself, 
I venture to assert thatdt never makes any real addition 
to our knowledge, in the meaning of the expression some- 
times accepted. As in other cases of inference, it merely 
unfolds the information contained in past observations; 
it merely renders explicit v.'hat was implicit in previous 
experience. It transmutes, but certainly does not create 

There is no fact which I shall more constantly keep 
before the reader's mind in the following pages than that 
the results of imperfect induction, however well authen- 
ticated and verified, are never more than probable. Wo 
never can be sure that the future will be as the present. 
We hang ever upon the will of the Creator: and it is 
only so far as He lias created two things alike, or maintains 
the framework of the world unchanged from moment to 
moment, that our most careful inferences can be fulfilled. 
All predictions, all inferences which reach beyond their 
data, are purely hypothetical, and proceed on the assump- 








tion that new events will conform to the conditions detected 
in our observation of past events. No experience of finite 
duration can give an exhaustive knowledge of the forces 
which are in operation. There is thus a double uncertainty • 
even supposing the Universe as a whole to proceed un- 
changed, we do not really know tlie Universe as a wliole 
We know only a point in its infinite extent, and a moment 
m Its infinite duration. We cannot bo sure, then, that our 
observations have not escaped some fact, which will cause 
the future to be apparently different from the past • nor 
can we be sure that the future really will be the outcome 
of the past. We proceed then in all our inferences to 
unexamined objects and times on the assumptions 

1. That our past observation gives us a complete know- 

ledge of what exists. 

2. That the conditions of things which did exist 

will continue to be the conditions which will 

We shall often need to illustrate the character of our 
knowledge of nature by the simile of a ballot-box, so often 
employed by mathematical writera in the theory of proba- 
bility. Nature is to us like an infinite ballot-box the 
contents of which are being continually drawn, ball after 
ball, and exhibited to us. Science is but the careful 
observation of the succession in which balls of various 
character pi-eseiit themselves; we register the combina- 
tions, notice those which seem to be excluded from occur- 
rence, and from the proportional frequency of those which 
appear we infer the probable character of future drawings 
But under such circumstances certainty of prediction 
depends on two conditions : — 

I. Thai we acquire a perfect knowledge of the com- 

parative numbei-s of balls of each kind within 

the box. 

2^ That the contents of the ballot-box remain unchanrred 
Of the latter assumption, or rather that conceniinc? the 
constitution of the world which it illustrates, the lo«?ician 
or physicist can have nothing to say. As the Creatfon of 
the Universe IS necessarily an act passing aU experience 
and all conception, so any change in that Universe or it 
may be, a termination of it, must likewise be infinitely be- 
yond the bounds of our mental faculties. No 8cien(4 no 




reasoning upon the subject, can have any validity; for 
without experience we are without the basis and materials 
of knowledge. It is the fundamental postulate accordingly 
of all inference concerning the future, that there shall be 
no arbitrary change in the subject of inference ; of the pro- 
bability or improbability of such a change I conceive that 
our faculties can give no estimate. 

The other condition of inductive inference — that we 
acquire an approximately complete knowledge of the com- 
binations in which events do occur, is in some degree 
within our power. There are branches of science in which 
phenomena seem to be governed by conditions of a most 
fixed and general character. We have ground in such 
cases for believing that the future occurrence of such 
phenomena can be calculated and predicted. But the 
whole question now becomes one of probability and im- 
probability. We seem to leave the region of logic to enter 
one in which the number of events is the ground of in- 
ference. We do not really leave the region of logic ; we 
only leave that where certainty, affirmative or negative, is 
the i-esult, and the agieement or disagreement of qualities 
the means of inference. For the future, number and 
quantity will commonly enter into our processes of reason- 
ing ; but then I hold that number and quantity are but 
portions of the great logical domain. I venture to assert 
that number is wholly logical, both in its fundamental 
nature and in its developments. Quantity in all its forms 
is but a development of number. That which is mathe- 
matical is nut the less logical ; if anything it is more 
logical, in the sense that it presents logical results in a 
higher degree of complexity and variety. 

Before proceeding then from Perfect to Imperfect In- 
duction I must devote a portion of this work to treating 
the logical conditions of number. I shall then employ 
number to estimate the variety of combinations in which 
natural phenomena may present themselves, and the pro- 
bability or improbability of their occurrence under definite 
circumstances. It is in later parts of the work that I must 
endeavour to establish the notions which I have set forth 
upon the subject of Imperfect Induction, as applied in the 
investigation of Nature, which notions may be thus briefly 
stated : — 

1 1 



1. Imperfect Induction entirely rests upon Perfect In- 

duction for its materials. 

2. The logical process by which we seem to pass directly 

from examined to unexamined cases consists in an 
inverse application of deductive inference, so that 
all reasoning may be said to be either directly or 
inversely deductive. 

3. The result is always of a hypothetical character, and 

is never more than probable. 

4. No net addition is ever made to our knowledge by 

reasoning ; what we know of future evente or un- 
examined objects is only the unfolded contents of 
our previous knowledge, and it becomes less pro- 
bable as it is more boldly extended to remote 







Not without reason did Pythagoras represent the world 
as ruled by number. Into almost all our acts of thought 
number enters, and in proportion as we can define numeri- 
caUy we enjoy exact and useful knowledge of the Universe 
The science of numbers, too, has hitherto presented the 
widest and most practicable training in logic. So free and 
energetic has been the study of mathematical forms, com- 
pared with the forms of logic, that mathematicians have 
passed far m advance of pure logicians. Occasionally, in 
recent times, they have condescended to apply their 
algebraic instrument to a reflex treatment of the primary 
logical science. It is thus that we owe to profound mathe- 
maticians, such as John Herschel, WheweU, De Morgan, or 
Boole, the regeneration of logic in the present century ' I 
entertain no doubt that it is in maintaining a close alliance 
with quantitative reasoning that we must look for further 
progress in our comprehension of quaUtative inference 

I cannot assent, indeed, to the common notion that 
certainty begins and ends with numerical determination. 
JNothmg IS more certain than logical truth. The law* of 
Identity and difference are the tests of all that is certain 










throughout the range of thought, and mathematical reason- 
ing is cogent only when it couforms to these conditions, of 
which logic is the first development And if it be 
erroneous to suppose that all certainty is mathematical, it 
is equally an error to imagine that all which is mathe- 
matical is certain. Many processes of mathematical 
reasoning are of most doubtful validity. There are points 
of mathematical doctrine which must long remain matter 
of opinion ; for instance, the best form of the definition and 
axiom concerning parallel lines, or the true nature of a 
limit. In the use of symbolic reasoning questions occur on 
which the best mathematicians may differ, as Bernoulli 
and Leibnitz differed irreconcileably concerning the exis- 
tence of the logarithms of negative quantities.^ In fact we 
no sooner leave the simple logical conditions of number, 
than we find ourselves involved in a mazy and mysterious 
science of symbols. 

Mathematical science enjoys no monopoly, and not even 
a supremacy, in certainty of results. It is the boundless 
extent and variety of quantitative questions that delights 
the mathematical student When simple logic can give 
but a bare answer Yes or No, the algebraist raises a score 
of subtle questions, and brings out a crowd of curious 
results. The flower and the fmit, all that is attractive 
and delightful, fall to the share of tlie mathematician, who 
too often despises the plain but necessary stem from which 
all has arisen. In no region of thought can a reasoner 
cast himself free from the prior conditions of logical cor- 
rectness. The mathematician is only strong and true as 
long as he is logical, and if number rules the world, it is 
logic which rules number. 

Nearly all writers have hitherto been strangely content 
to look upon numerical reasoning as something apart from 
logical inference. A long divorce has existed between 
quality and quantity, and it has not been uncommon to 
treat them as contrasted in nature and restricted to 
independent branches of thought For my own part, I 
believe that all the sciences meet somewhere. No part of 
knowledge can stand wholly disconnected from other parts 
of the universe of thought ; it is incredible, above all, that 

' MoQtucla, Uistoire dc* MaUUmaiiquet, vol. iii. p. 373. 




the two great branches of abstract science, interlacing and 
co-operating in every discourse, should rest upon totally 
distinct foundations. I assume that a connection exists, 
and care only to inquire. What is its nature ? Does the 
science of quantity rest upon that of quality; or, vice 
versd, does the science of quality rest upon that of 
quantity? There might conceivably be a third view, 
that tbey both rest upon some still deeper set of prin- 

It is generally suj>posed that Boole adopted the second 
view, and treated logic as an application of algebra, a 
special case of analytical reasoning which admits only two 
quantities, unity and zero. It is not easy to ascertain 
clearly which of these views really was accepted by Boole. 
In his interesting biographical sketch of Boole,^ the Eev. 
K. Harley protests against the statement that Boole's 
logical calculus imported the conditions of number and 
quantity into logic. He says : « Logic is never identified 
or confounded with mathematics; the two systems of 
thought are kept perfectly distinct, each being subject to 
its own laws and conditions. The symbols are the same 
for both systems, but they have not the same intei-pre- 
tatiou." The Eev. J. Venn, again, in his review of Boole's 
logical system,2 holds that Boole's processes are at bottom 
logical, not mathematical, though stated in a highly gener- 
alized fonn and with a mathematical dress. But it is 
quite likely that readers of Boole should be misled. Not 
only have his logical works an entirely mathematical 
appearance, but I find on p. 12 of his Laws of TJioufjht 
the following unequivocal statement: "That logic, as a 
science, is susceptible of very wide applications is 
admitted; but it is equally certain that its ultimate 
forms and processes are mathematical" A few lines 
below he adds, " It is not of the essence of mathematics 
to be conversant with the ideas of number and quantity." 

The solution of the difficulty is that Boole used the 
terna mathematics in a wider sense than that usually 
attributed to it He pi-obably adopted the third view, so 
that his mathematical Laws of Thought are the common 

» British Quarterly Review, No. Ixxxvii, July 1866. 
' Mind; October 1876, vol. i. p. 484. 





basis both of logic and of quantitative mathematics. But 
I do not care to pursue the subject because I think that 
in either case Boole was wrong. In my opinion logic is 
the superior science, the general basis of mathematics as 
well as of all other sciences. Number is but logical dis- 
crimination, and algebra a highly developed logic. Thus 
it is easy to understand the deep analogy which Boole 
pointed out between the fonns of algebraic and logical 
deduction. Logic resembles algebra as the mould 
resembles that which is cast in it Boole mistook the 
cast for the mould. Considering that logic imposes its 
own laws upon every branch of mathematical science, it 
is no wonder that we constantly meet with the traces of 
logical laws in mathematical processes. 

The Nature of Number. 

Number is but another name for diversity. Exact iden- 
tity is unity, and with difference arises plurality. An 
abstract notion, as was pointed out (p. 28), possesses a 
certain oneness. The quality of fustice, for instance, is one 
and the same in whatever just acts it is manifested. In 
justice itself there are no marks of difference by which to 
discriminate justice from justice. But one just act can be 
discriminated from another just act by circumstances of 
time and place, and we can count many acts thus discri- 
minated each from each. In like manner pure gold is 
simply pure gold, and is so far one and the same through- 
out. But besides its intrinsic qualities, gold occupies 
space and muse have shape and size. Poi-tions of gold 
are always mutually exclusive and capable of discrimina- 
tion, in respect that they must be each without the other. 
Hence they may be numbered. 

Plurality arises when and only when we detect differ- 
ence. For instance, in counting a number of gold coins 
I must count each coin once, and not more than once. 
Let C denote a coin, and the mark above it the order of 
counting. Then I must count the coins 

C + C" + C" + C" + 

If I were to count them as follows 

C + C + C"' + C" + CT'' + . . ., 
I should make the tliird coin into two, and should imply 




the existence of difference where there is no difference.* 
C" and C' are but the names of one coin named twice 
over. But according to one of the conditions of logical 
symbols, which I have called the Law of Unity (p. 72), 
the same name repeated has no effect, and 

A + A = A. 
We must apply the Law of Unity, and must reduce all 
identical alternatives before we can count with certainty 
and use the processes of numerical calculation. Identical 
alternatives are harmless in logic, but are wholly inad- 
missible in number. Thus logical science ascertains the 
nature of the mathematical unit, and the definition may 
be given in these terms — A unit is any object of thought 
which can he discriminated from every other object treated as 
a unit in the same problem. 

It has often been said that units are unfts in respect of 
being perfectly similar to each other ; but though they 
may be perfectly similar in some respects, they must be 
different in at least one point, otherwise they would be 
incapable of plurality. If three coins were so similar that 
they occupied the same space at the same time, they 
would not be three coins, but one coin. It is a property 
of space that every point is discriminable from every other 
point, and in time every moment is necessarily distinct 
from any other moment before or after. Hence we 
frequently count in space or time, and Locke, with some 
other philosophers, has held that number arises from 
repetition in time. Beats of a pendulum may be so 
perfectly similar that we can discover no difference except 
that one beat is before and another after. Time alone is 
here the ground of difference and is a sufficient foundation 
for the discrimination of plurality ; but it is by no means 
the only foundation. Three coins are three coins, whether 
we count them successively or regard them all simul- 
taneously. In many cases neither time nor space is the 
ground of difference, but pure quality alone enters. We 
can discriminate the weight, inertia, and hardness of gold 
as three qualities, though none of these is before nor after 
the other, neither in space nor tima Every means of 
discrimination may be a source of plurality. 

* Pwr€ Logic, Appendix, p. 82, \ 192 







'V (■ 



Our logical notation may be used to express the rise of 
number. The symbol A stands for one thing or one class, 
and in itself must be regarded as a unit, because no 
difference is specified. But the combinations AB and Ah 
are necesssarily two, because they cannot logically coalesce, 
and there is a mark B which distinguishes one from the 
other. A logical definition of the number four is given in 
the combinations ABC, ABc, AhC, Ahc, where there is a 
double difference. As Puck says — 

" Yet but three ? Come one more ; 
Two of both kinds makes up four." 

I conceive that all numl)ers might be represented as 
arismg out of the combinations of the logical Alphabet, 
more or less of each series being struck out by various 
logical conditions. The number three, for instance, arises 
from the condition that A must be either B or C, so that 
the combinations are ABC, ABc, AbC. 

0/ Numerical Abstraction. 

Tliere will now be little difficulty in forming a clear 
notion of the nature of numerical abstraction. It consists 
in abstracting the character of the difference from which 
plurality anses, retaining merely the fact. When I speak 
ot three men I need not at once specify the marks by which 
each may be known from each. Those marks must exist 
if they are really three men and not one and the same and 
m speaking of them as many I imply the existence of the 
requisite differences. Abstract number, then, is iJie empty 
form of difference ; the abstract number three asserts the ex- 
istence of marks without specifying their kind. 

Numerical abstraction is thus seen to be a dif- 
ferent process from logical abstraction (p. 27), for in the 
latter process we drop out of notice the very existence of 
difference and pluraUty. In forming the abstract notion 
hardriess we ignore entirely the diverse circumstances in 
which the quality may appear. It is the concrete notion 
three hard objects, which asserts the existence of hardness 
along with sufficient other undefined qualities, to mark out 
three such objects. Numerical thought is indeed closely 
interwoven with logical thought. We cannot use a con 

Crete term in the plural, as men, without implying that 
there are marks of difference. But when we use an 
abstract term, we deal with unity. 

The origin of the great generality of number is now 
apparent. Three sounds differ from three colours, or three 
riders from three horses ; but they agree in respect of the 
variety of marks by which they can be discriminated. The 
symbols 1+1+ 1 are thus the empty marks asserting the 
existence of discrimination. But in dropping out of sight 
the character of the differences we give rise to new 
agreements on which mathematical reasoning is founded. 
Numerical abstraction is so far from being incompatible 
with logical abstraction that it is tlie origin of our widest 
acts of generalization. 

Concrete and Abstract Numher. 

The common distinction between concrete and abstract 
number can now be easily stated. In proportion as we 
specify the logical characters of the things numbered, we 
render them concrete. In the abstract number thru 
there is no statement of the points in which the three 
objects agree ; but in three coins, three men, or three Jwrses, 
not only are the objects numbered but their nature is re- 
stricted. Concrete number thus implies^ the same con- 
sciousness of difference as abstract number, but it is 
mingled with a groundwork of similarity expressed in the 
logical terms. There is identity so far as logical terms 
enter ; difference so far as the terms are merely numerical. 

The reason of the important I^w of Homogeneity will 
now be apparent. This law asserts that in every arith- 
metical calculation the logical nature of the things num- 
bered must remain unaltered. The specified logical 
agreement of the things must not be affected by the un- 
specified numericsd differences. A calculation would be 
palpably absurd which, after commencing with length, 
gave a result in hours. It is equally absurd, in a purely 
arithmetical point of view, to deduce areas from the 
calculation of lengths, masses from the combination of 
volume and density, or momenta from mass and velocity. 
It must remain for subsequent consideration to decide in 
what sense we may truly say that two linear feet multi- 








plied by two linear feet give four superficial feet ; arith- 
metically it is absurd, because there is a change of unit. 

As a general rule we treat in each calculation only 
objects of one natura We do not, and cannot properly 
add, in the same sum yards of cloth and pounds of sugar 
We cannot even conceive the result of adding area to 
velocity, or length to density, or weight to value. The 
units added must have a basis of homogeneity, or must be 
reducible to some common denominator. Nevertheless it 
is possible, and in fact common, to treat in one complex 
calculation the most heterogeneous quantities, on the 
condition that each kind of object is kept distinct, and 
treated numerically only in conjunction with its own kind. 
Different units, so far as their logical differences are speci- 
fied, must never be substituted one for the other. Chemists 
continually use equations which assert the equivalence of 
groups of atoms. Ordinary fermentation is represented 
by the fornmla 

C* H" 0* = 20* H« O + 200«. 

Three kinds of units, the atoms respectively of carbon, 
hydrogen, and oxygen, are here intermingled, but there is 
really a separate equation in regard to each kind. Mathe- 
maticians also employ compound equations of the same 
kind ; for in, a + J v/ - I = c 4- ^ v/ - I, it is impossible 
by ordinary addition to add atohy/— i. Hence we 
really have the separate equations a = b, and c ij — i = d 
J — I. Similarly an equation between two quaternions is 
equivalent to four equations between ordinary quantities, 
whence indeed the name quaternion. 

Analogy of Logical and Numtrical Terms. 

If my assertion is correct that number arises out of 
logical conditions, we ought to find number obeying all the 
laws of logic. It is almost superfluous to point out that 
this is the case with the fundamental laws of identity and 
difference, and it only remains to show that mathematical 
symbols do really obey the special conditions of logical 
symbols which were formerly pointed out (p. 32). Thus 
the Law of Oommutativeness, is equally true of quality and 
quantity. As in logic we have 

AB = BA, 
80 in mathematics it is familiarly known that 

2x3 = 3x2, or X X If =^y X X. 
The properties of space are as indifferent in multiplication 
as we found them m pure logical thought. 
Similarly, as in logic 

triangle or square = square or triangle 
or generally A + B = B .|. A, • 

80 in quantity 2 + 3 = 3 + 2' 

or generally x -^ y = y + x. 

The symbol f. is not identical with +, but it is thus far 

How far now, is it true that mathematical symbols obey 
the Law of Simplicity expressed in the form 

AA = A, 
or the example 

Round round = round ? 
Apparently there are but two numbers which obey this 
law ; for it is certain that "^ 

XXX ^s v 

is true only in the two cases when a; = i, or a; = o 

In reality all numbers obey the law, for 2 x 2 ='2 is not 
really analogous to AA = A. According to the definition 
of a unit already given, each unit is discriminated from 
each other m the same problem, so that in 2' x 2" the 
first two involves a different discrimination from the second 
two. I get four kinds of things, for instance, if I first dis- 
criminate •' heavy and light" and then "cubical and 
spherical, for we now have the foUowing classes- 
heavy, cubical light, cubical 
heavy, spherical. light, spherical 
But suppose that my two classes are in both cases dis- 
weT^ve ^^™^ difference of light and heavy then 

heavy heavy = heavy, 

heavy light = o, 

light heavy = o, 

Hght light = light 
Ihus, Oieavy or light) x (heavy or light) = (heavy or light). 
In short, twice two ts two unless we take care that the 
second two has a different meaning from the first. But 
l.nH ^'"^"^I circumstances logical terms give the like 
result, and it is not true that A'A" = A', when A" is 
different in meaning from A'. 












In a similar manner it may be shown that the Law of 
Unity A i A = A. 

holds true alike of logical and mathematical terms. It is 
absurd indeed to say that 

except in the one case when x = absolute zero. But this 
contradiction a; + a; = a; arises from the fact that we have 
already defined the units in one x as differing from those in 
the other. Under such circumstances tlie Law of Unity 

does not apply. For if in 

A' + A-' -A' 
we mean that A" is in any way different from A' the 
assertion of identity is evidently false. 

The contrast then which seems to exist between logical 
and mathematical symbols is only apparent It is because 
the Laws of Simplicity and Unity must always be observed 
in the operation of counting that those laws s6em no further 
to apply. This is the understood condition under which 
we use all numerical symbols. Whenever I wnte the 
symbol 5 I really mean 

I + I 4 I + I + I, . . 

and it is perfectly understood that each of these units is 
distiuct from each other. If requisite I might mark them 

"'+ !'"'+ i'"". 

i'+ r + I 

Were this not the case and were the units really 

I' + I" + I" + I'" + I"", 
the Law of Unity would, as before remarked, apply, and 

l" 4- I" = I". 

Mathematical symbols then obey all the laws of logical 
symbols, but two of these laws seem to be inapplicable 
simply because they are presupposed in the definition of 
the mathematical unit. Logic thus lays down the con- 
ditions of number, and the science of arithmetic developed 
as it is into all the wondrous branches of mathematical 
calculus is but an outgrowth of logical discrimination. 

Principle of Mathematical Inference. 

The universal principle of all reasoning, as I have 
asserted, is that which allows us to substitute like for like. 
I have now to point out how in the mathematical sciences 

this principle is involved in each step of reasoning. It is 
m these sciences indeed that we meet with the clearest 
cases of substitution, and it is the simplicity with which 
the principle can be applied which probably led to the 
compamtively early perfection of the sciences of geometry 
and arithmetic. Euclid, and the Greek mathematicians 
from the first, recognised equality as the fundamental 
relation of quantitative thought, but Aristotle rejected the 
exactly analogous, but far more general relation of identity 
and thus crippled the formal science of logic as it has 
descended to the present day. 

^ Geometrical reasoning starts from the axiom that 
"things equal to the same thing are equal to each other " 
Two cquahties enable us to infer a third equality ; and this 
IS true not only of lines and angles, but of areas, volumes, 
numbers, intervals of time, forces, velocities, degrees of 
intensity, or, in short, anything which is capable of being 
equal or unequal. Two stars equally bright with the same 
star must be equally bright with each other, and two forces 
equally intense with a third force are equally intense with 
each other. It is remarkable that Euclid has not explicitly 
stated two other axioms, the truth of which is necessarily 
^^ • l^* "^^^ second axiom should be that " Two things of 
which one is equal and the other unequal to a third com- 
mon thing, are unequal to each other." An equality and 
inequality, in short, give an inequality, and this is equaUy 
true with the first axiom of all kinds of quantity. If 
Venus, for instance, agrees with Mars in density, but Mars 
differs from Jupiter, then Venus differs from Jupiter. A 
third axiom must exist to the effect that " Things unequal 
to the same thing may or may not be equal to each 
other. Two inequalities give no ground of inference wJcat- 
wer. If we only know, for instance, that Mercury and 
Jupiter differ m density from Mars, we cannot say whether 
or not they agree between themselves. As a fact they do 
not agree ; but Venus and Mars on the other hand both 
differ from Jupiter and yet closely agree with each other. 
Ihe force of the axioms can be most clearly illustrated by 
drawing equal and unequal lines.* 

««1 '?^^^«'^ iwwTW tn Logui (Macmillan), p. 123. It is pointed 

w^ J^ ^i- r^w ^i^^l* ^'^"^ ^^'^""^ thit'^the views her? gfv^ 
were partially stated by Leibnitx, ^ 

M 2 









' I 

The general conclusion then must be that where there 
is equality there may be inference, but where there is not 
equality there cannot be inference. A plain induction 
will lead us to believe that eqicality is tlu condition of 
inference concerning quantity. All the three axioms may 
in fact be summed up in one, to the effect, that "tn 
whatever relation one quantity stands to another, it stands 
in the same relation to tJie equal of that other" 

The active power is always the substitution of equals, 
and it is an accident that in a pair of equalities we can 
make the substitution in two ways. From a = 6 = c we 
can infer a = c, either by substituting in a = b the value 
of 6 as given in b = c, or else by substituting in J = c the 
value of b as given in a = b. In a = 6 *« rf we can make 
but the one substitution of a for J. In «-'/-' </ we can 
make no substitution and get no inference. 

In mathematics the relations in which terms may stand 
to each other are for more varied than in pure logic, yet 
our principle of substitution always holds true. We may 
say in the most general manner that In whatever relation 
one quantity stands to another, it stands in the same relation 
to the equal of that other. In this axiom we sum up a 
number of axioms which have been stated in more or less 
detail by algebraists. Tlius, " If equal quantities be added 
to equal quantities, the sums will be equal." To explain 
this, let 

Now a -\- c, whatever it means, must be identical with 
itself, so that 

a + c = a •\' e. 
In one side of this equation substitute for the quantities 
their equivalents, and we have the axiom proved 

a + c = 6 + (£. 
The similar axiom concerning subtraction is equally evi- 
dent, for whatever a — c may mean it is equal to a — c, 
' and therefore by substitution Xx> b — d. Again, " if equal 
quantities be multiplied by the same op equal quantities, 
the products will be equal" For evidently 

a4i = ac^ 
and if for c in one side we substitute its equal d, we have 

ac = ad, 
and a second similar substitution gives us 

ac = hd. 
We might prove a like axiom concerning division in afi 
exactly similar manner. I might even extend the list of 
axioms and say that " Equal powers of equal numbers arc 
equal." For certainly, whatever ay ax a may mean, it 
is equal to a x a x « ; hence bv our usual substitution it 
IS equal to bxbxb. The same will be true of roots of 
numbers and IJa = *Jb provided that the roots are so 
taken that the root of a shall really be related to a as 
the root of b is to b. The ambiguity of meaning of an 
an operation thus fails in any way to shake the univ'ersality 
of the principle. We may go further and assert that, not 
only the above common relations, but all other known or 
conceivable mathematical relations obey the same prin- 
ciple. Let Qa denote in the most general manner that we 
do something with the quantity a ; then if a = 5 it follows 

Q« = QJ. 

The reader will also remember that one of the most 
frequent operations in mathematical reasoning is to sub- 
stitute for a quantity its equal, as known either by assumed, 
natural, or self-evident conditions. Whenever a quantity 
appears twice over in a problem, we may apply what we 
learn of its relations in one place to its relations in the 
other. All reasoning in mathematics, as in other branches 
of science, thus involves the principle of treating equals 
equally, or similars similarly. In whatever way we 
employ quantitative reasoning in the remaining parts of 
this work, we never can desert the simple principle on 
which we first set out i^ r 

Reasoning by Inequalities. 

I have stated that all the processes of mathematical 
reasoning may be deduced from the principle of substi- 
tution. Exceptions to this assertion may seem to exist 
in the use of inequalities. The greater of a greater is 
undoubtedly a greater, and what is less than a less is 
certainly less. Snowdon is higher than the Wrekin, and 
Ben Nevis than Snowdon ; therefore Ben Nevis is higher 
than the Wrekin. But a little consideration discloses 
*»ufficient reason for believincr that even in such cases, 











where equality does not apparently enter, the force of the 
reasoning entirely depends upon underlying and implied 

In the first place, two statements of mere difference do 
not give any ground of inference. We Jeam nothing 
concerning the comparative heights of St. Paul's and 
Westminster Abhey from the assertions that they both 
differ in height from St. Peter's at Kome. We need some- 
thing more than inequality ; we require one identity in 
addition, namely the identity in direction of the two 
differences. Thus we cannot employ inequalities in the 
simple way in which we do equalities, and, when we try 
to express what other conditions are requisite, we find 
ourselves lapsing into the use of equalities or identities. 

In the second place, every argument by inequalities 
may be represented in the form of equalities. We express 
that a is greater than h by the equation 

a = 2>+jp, (l) 

where p is an intrinsically positive quantity, denoting the 
difference of a and b. Similarly we express that b is 
greater than c by the equation 

6 = c + g, (2) 

and substituting for 6 in (i) its value in (2) we have 

a==c + q+p. (3) 

Now as p and q are both positive, it follows that a is 
greater than c, and we have the exact amount of excess 
specified. It will be easily seen that the reasoning con- 
cerning that which is less than a less will result in an 
equation of the form 

c z= a -- r " 8. 
Every argument by inequalities may then be thrown 
into the form of an equality ; but the converse is not true. 
We cannot possibly prove that two quantities are equal 
by merely asserting that they are both greater or both le^s 
than another quantity. From e >f and ^ >/, or e <f 
and g </,vfe can infer no relation between e and g. And 
if the reader take the equations a? = y = 3 and attempt to 
prove that therefore a; = 3, by throwing them into in- 
equalities, he will find it impossible to do so. 

From these considerations I gather that reasoning in 
arithmetic or algebra by so-called inequalities, is only an 
imperfectly expressed reasoning by equalities, and when 

we want to exhibit exactly and clearly the conditions of 
reasoning, we are obliged to use equalities explicitly. Just 
as in pure logic a negative proposition, as expressing mere 
difference, cannot be the means of inference, so inequalitv 
can never really be the true ground of inference. I do 
not deny that affirmation and negation, agreement and 
difference, equality and inequality, are pairs of equally 
fundamental relations, but I assert that inference is pos- 
sible only where affirmation, agreement, or equality, some 
species of identity in fact, is present, explicitly or implicitly. 

Arithmetical Reasoning, 

It may seem somewhat inconsistent that I assert number 
to arise out of difference or discrimination, and yet hold 
that no reasoning can be grounded on difference. Number 
of course, opens a most wide sphere for inference, and a 
bttle consideration shows that this is due to the unlimited 
senes of identities which spring up out of numerical 
abstraction. If six people are sitting on six chairs, there 
is no resemblance between the chairs and the people in 
logical character. But if we overlook all the qualities 
both of a chair and a person and merely remember that 
there are marks by which each of six chairs may be 
discriminated from the othei-s, and similarly with the 
people, then there arises a resemblance between the chairs 
and the people, and this resemblance in number may be 
the ground of inference. If on another occasion the chairs 
are hi ed by people again, we may infer that these people 
resemble the othera in number though they need not 
resemble them in any other points. 

Groups of units are what we really treat in arithmetic. 
The number Jive is really i + i + i + i + i, but for the 
sake of conciseness we substitute the more compact sign 
5, or the name Jive. These names being arbitrarily im- 
posed m any one manner, an infinite variety of relations 
spring up between them which are not in the least 
arbitrary. If we define fonr as I + i + i + i, and >e 
as 1 + I + 1 + I + I, then of course it follows that 
five =>wr + I ; but it would be equally possible to take 
this latter equality as a definition, in which case one of 
tne former equalities would become an inference It is 






hardly requisite to decide how we define the names of 
numbers, provided we remember that out of the infinitely 
numerous relations of one number to others, some one 
relation expressed in an equality must be a definition of 
the number in question and the other relations imme- 
diately become necessary inferences. 

In the science of number the variety of classes which 
can be formed is altogether infinite, and statements of 
perfect generality may be made subject only to difficulty 
or exception at the lower end of the scale. Every existing 
number for instance belongs to the class m + 2; that is, 
every number must be the sum of another number and 
seven, except of coui-se the first six or seven numbers, 
negative quantities not being here taken into account. 
Every number is the half of some other, and so on. The 
subject of generalization, as exhibited in mathematical 
truths, is an infinitely wide one. In number we are only 
at the first step of an extensive series of generalizations. 
As number is general compared with the particular things 
numbered, so we have general symbols for numbers, and 
general symbols for relations between undetermined 
numbei*s. There is an unlimited hierarchy of successive 

Numerically Definite Reasoning, 

It was first discovered by De Morgan that many argu- 
ments are valid which combine logical and numerical 
reasoning, although they cannot ^ included in the 
ancient logical formulas. He developed the doctrine of 
the " Numerically Definite Syllogism,** fully explained in 
his Formal Logic (pp. 141 — 170). Boole also devoted 
considerable attention to the detennination of what he 
called "Statistical Conditions," meaning the numerical 
conditions of logical classes. In a paper published among 
the Memoirs of the Manchester Literary and Pliilosophical 
Society, Third Series, voL IV. p. 330 (Session 1869—70), 
I have p* inted out that we can apply arithmetical calcula- 
tion to the Logical Alphabet. Having given certain logical 
conditions and the numbers of objects in certain classes, 
we can either determine the numbers of objects in other 
classes governed by those conditions, or can show what 




further dat^a are required to determine them. As an 
example of the kind of questions treated in numerical 
logic, and the mode of treatment, I give the following 
problem suggested by De Morgan, with my mode of 
representing its solution. 

'* For every man in the house there is a person who is 
aged ; some of the men are not aged. It follows that 
some persons in the house are not men."^ 
Now let A = person in house, 

B = male, 
C = aged. 
By enclosing a logical symbol in brackets, let us denote 
the number of objects belonging to the class indicated by 
the syn^bol. Thus let 

(A) = number of persons in house, 
(AB) = number of male persons in house, 
(ABC) ■= number of aged male persons in house, 
and so on. Now if we use w and w' to denote unknown 
numbers, the conditions of the problem may be thus stated 
according to my interpretation of the words — 

tliat IS to say, the number of persons in the house who are 
aged is at least equal to, and may exceed, the number of 
male persons in the house ; 

(ABc) = < (2) 

that is to say, the number of male persons in the house 
who are not aged is some unknown positive quantity. 

If we develop the terms in (i) by the Law of Duality 
(pp. 74, 81, 89), we obtain 

(ABC) + (ABc) = (ABC) + (AJC) - w, 
Subtractmg the common term (ABC) from each side and 
substituting for (ABc) its value as given in (2), we get at 

(A5C) = w-^w\ 
and adding {Abe) to each side, we have 

(A6) = {Abe) -{■w + w\ 
Ihe meanmg of this result is that the number of persons 
in the house who are not men is at least equal io w + w\ 
and exceeds it by the number of persons in the house who 
are neither men nor aged (AJc). 

* Sylladui of a Proposed SysUm of Logic, p. 29. 









It should be understood that this solution applies only 
to the terms of the example quoted above, and not to the 
general problem for wliich De Morgan intended it to serve 
as an illustration. 

As a second instance, let us take the following ques- 
tion : — The whole number of voters in a borough is a ; 
the number against whom objections have been lodged by 
liberals is h; and the number against whom objections 
have been lodged by conservatives is c; required the 
number, if any, who have been objected to on both sides. 

A = voter, 

B = objected to by liberals, 

C = objected to by conservatives, 

then we require the value of (ABC). Now the following 

equation is identically true — 

(ABC) = (AB) + (AC) + {Ahc) - (A). (i) 

For if we develop all the terms on the second side we 


^ABC) = (ABC) + (ABc) + (ABC) + (A6C) + (Abe) 
- (ABC) - (ABc) - (A6C) - {Khc) ; 
and striking out the corresponding positive and negative 
terras, we have left only (ABC) = (ABC). Since then 
(i) is necessarily tme, we have only to insert the known 
values, and we have 

(ABC) =.h-\-c-a-\- (A6c). 
Hence the number wlio have received objections from both 
sides is equal to the excess, if any, of the whole number 
of objections over the number of voters together with the 
number of voters who have received no objection {Xhc). 

The following problem illustrates the expression for 
the common part of any three classes: — The number of 
paupers who are blind males, is equal to the excess, if 
any, of the sum of the whole number of blind persons, 
added to the whole number of male persons, added to the 
number of those who being paupers are neither blind nor 
males, above the sum of the whole number of paupers 
added to the number of those who, not being paupers, 
are blind, and to the number of those who, not being 
paupers, are male. 

The reader is requested to prove the truth of the above 
statement (i) by his own unaided common sense; (2) by 

the Aristotelian Logic ; (3) by the method of numerical 
logic just expounded ; and then to decide which method 
is most satisfactory. 

Numeiical meaning of Logical Gonditicms. 

In many cases classes of objects may exist under spe- 
cial logical conditions, and we must consider how these 
conditions can be interpreted numerically. Every logical 
proposition gives rise to a corresponding numerical 
equation. Sameness of qualities occasions sameness of 
numbers. Hence if 

A = B 
denotes the identity of the qualities of A and B, we may 
conclude that 

(A) = (B). 
It is evident that exactly those objects, and those objects 
only, which are comprehended under A must be compre- 
hended under B. It follows that wherever we can draw 
an equation of qualities, we can di-aw a similar equation 
of numbers. Thus, from 

A = B = C 

we infer 


and similarly from 

(A) = (B) = (C), 
meanmg that the numbers of A's and C s are equal to the 
number of B's, we can infer 

. , (A) = (C). 

But, cunously enough, this does not apply to negative 
propositions and inequalities. For if 

A = B - D 
means that A is identical with B, which differs from D, it 
does not follow that 

(A) = (B) ^ (D). 
Two classes of objects may differ in qualities, and yet they 
niay agree in number. This point strongly confirms me 
in the opinion which I have already expressed, that all 
inference really depends upon equations, not differences. 

The Logical Alphabet thus enables us to make a com- 
plete analysis of any numerical problem, and though the 
symbolical statement may sometimes seem prolix, I con- 




ceive that it really represents the course which the mind 
must follow in solving the question. Although thought 
may outstrip the rapidity with which the symbols can 
be written down, yet the mind docs not really follow a 
different course from that indicated by the symbols. For 
a fuller explanation of this natural system of Numerically 
Definite Eeasoning, with more abundant illustrations 
and an analysis of De Morgan's Numerically Definite 
Syllogism, I must refer the reader to the paper^ in the 
Memoirs of the Manchester Literary and Philosophical 
Society, already mentioned, portions of which, however, 
have been embodied in the present section. 

The reader may be referred, also, to Boole's writin^^s 
upon the subject in the Laws of Thought, chap. xix. 
p. 295, and in a paper on "Propositions Numerically 
Definite," communicated by De Morgan, in 1868, to the 
Cambridge Philosophical Society, and printed in their 
Transactions;' vol. xi. part ii. 

» Jt has b«jn pointed out to me by Mr. C. J. Monroe, that section 14 
(P- 339) of this paper is erroneous, and oaj?ht to be cancelled. The 
problem concerning the number of paupers illustrates the answer 
which should have been obtained. Mr. A. J. Ellis, F.R.S., ha<l 
previously observed that my solution in the paper of De Morgan's 
problem about " men in the house " did not answer the conditions 
mtended by De Morgan, and I therefore give in the text a more 
satisfactory solution. 




Nature may be said to be evolved from the monotony 
of non-existence by the creation of diversity. It is plau- 
sibly asserted that we are conscious only so far as we 
experience difference. Life is change, and perfectly uni- 
form existence would be no better than non-existence. 
Certain it is that life demands incessant novelty, and that 
nature, though it probably never fails to obey the same 
fixed laws, yet presents to us an apparently unlimited 
series of varied combinations of events. It is the work of 
science to observe and record the kinds and comparative 
numbers of such combinations of phenomena, occurring 
spontaneously or produced by our interference. Patient 
and skilful examination of the records may then disclose 
the laws imposed on matter at its creation, and enable us 
more or less successfully to predict, or even to regulate, 
the future occurrence of any particular combination. 

The Laws of Thought are the first and most important 
of all the laws which govern the combinations of pheno- 
mena, and, though they be binding on the mind, they 
may aLso be regarded as verified in the external world. 
The Logical Alphabet develops the utmost variety of 
things and events which may occur, and it is evident that 
as each new quality is introduced, the number of combi- 
nations is doubled. Thus four qualities may occur in 16 
combinations; five qualities in 32; six qualities in 64; 
and 80 on. In general language, if n be the number of 
qualities, 2" is the number of varieties of things which 






may be fonned from them, if there be no conditions but 
those of logic. This number, it need hardly be said, 
increases after the first few terms, in an extraordinary 
manner, so that it would require 302 figures to express 
the number of combinations in which I,CXX) qualities 
might conceivably present themselves. 

If all the combinations allowed by the Laws of Thought 
occuned indifferently in nature, then science would begin 
and end with those laws. To observe nature would give 
us no additional knowledge, because no two qualities 
would in the long run be oftener associated than any 
other two. We could never predict events with more 
certainty than we now predict the throws of dice, and 
experience would be without usa But the universe, as 
actually created, presents a far different and much more 
interesting problem. The most superficial observation 
shows that some things are constantly associated with 
other things. The more mature our examination, the 
more we become convinced that each event depends 
upon the prior occurrence of some other series of events. 
Action and reaction are gradually discovered to underlie 
the whole scene, and an independent or casual occurrence 
does not exist except in appearance. Even dice as they 
fall are surely determined in their course by prior con- 
ditions and fixed laws. Thus the combinations of events 
which can really occur are found to be comparatively 
restricted, and it is the work of science to detect these 
restricting conditions. 

In the English alphabet, for instance, we have twenty- 
six letters. Were the combinations of such letters per- 
fectly free, so that any letter coidd be indifferently 
sounded with any other, the number of words which 
could be formed without any repetition would be 2^ — i, 
or 67,108,863, equal in number to the combinations of 
the twenty-seventh column of the Logical Alphabet, 
excluding one for the case in which all the letters 
would be absent. But the formation of our vocal 
organs prevents us from using the far greater part of 
these conjunctions of letters. At least one vowel must b*» 
present in each word ; more than two consonants cannot 
usually be brought together ; and to produce words capable 
of smooth utterance a number of other rules must be 



observed. To determine exactly how many words might 
exist in the English language under these circumstances, 
would be an exceedingly complex problem, the solution of 
which has never been attempted. The number of existing 
English words may perhaps be said not to exceed one 
hundred thousand, and it is only by investigating the com- 
binations presented in the dictionary, that we can learn the 
Laws of Euphony or calculate the possible number of 
words. In this example we have an epitome of the work 
and method of science. The combinations of natural 
phenomena are limited by a great number of conditions 
which are in no way brought to our knowledge except so 
far as they are disclosed in the examination of nature. 

It is often a very difficult matter to determine the num- 
bers of permutations or combinations which may exist 
under various restrictions. Many learned men puzzled 
themselves in former centuiies over what were called 
Protean verses, or verses admitting many variations in 
accordaace with tlie Laws of Metre. The most celebrated 
of these verses was that invented by Bernard Bauhusius, 
as follows : ^ — 

** Tot tibi aiwt dotes, Virgo, quot sidera cceIo.* 

One author, Erioius Puteanus, filled forty-eight pages of a 
work in reckoning up its possible transpositions, making 
them only 1022. Other calculators gave 2196, 3276, 2580 
as their results. Wallis assigned 3096, but without much 
confidence in the accuracy of his result.^ It required the 
skill of James Bernoulli to decide that the number of 
transpositions was 3312, under the condition that the sense 
and metre of the verse shall be perfectly preserved. 

In approacliing the consideration of the great Inductive 
problem, it is very necessary that we sliould acquire correct 
notions as to the comparative numbers of combinations 
which may exist under different circumstances. The 
doctrine of combinations is that part of mathematical 
science which applies numerical calculation to determine 
the numbers of combinations under various conditions. 
It is a part of the science which really lies at the base not 
only of other sciences, but of other branches of mathe- 

J Montucla, Ritioirty &c., vol. iii. p. 388. 
• Wallis, Of CombinaiioMf &c., p. iiQt 





matics. The forms of algebraical expressions are deter- 
mined by the principles of combination, and Hindenburg 
recognised this fact in his Combinatorial Analysis. The 
greatest mathematicians have, during the last three cen- 
turies, given their best powers to the treatment of this 
subject ; it waa the favourite study of Pascal ; it early 
attracted the attention of Leibnitz, who wrote his curious 
essay, Be Arte Cmnbinatoria, at twenty years of age ; James 
Bernoulli, one of the very profoundest mathematicians, 
devoted no smaU part of his life to the investigation of the 
subject, as connected with that of Probability ; and in his 
celebrated work, Be Arte Gonjectandi, he has so finely 
described the importance of the doctrine of combinations, 
that I need offer no excuse for quoting his remarks at full 

" It is easy to perceive that the prodigious variety which 
appears both in the works of nature and in the actions of 
men, and which constitutes the greatest part of the beauty 
of the universe, is owing to the multitude of different ways 
in which its several parts are mixed with, or placed near, 
each other. But, because the number of causes that concur 
m producing a given event, or effect, is oftentimes so im- 
mensely great, and the causes themselves are so different 
one from another, that it is extremely difficult to reckon up 
all the different ways in which they may be arranged or 
combined together, it often happens that men, even of the 
best understandings and greatest circumspection, are guilty 
of that fault in reasoning which the writers on logic call 
tAe insufficient or imperfect enumeratian of parts or cases : 
insomuch that I will venture to assert, that this is the 
chief, and almost the only, source of the vast number of 
erroneous opinions, and those too very often in matters 
of great importance, which we are apt to form on all the 
subjects we reflect upon, whether they relate to the know- 
ledge of nature, or the merite and motives of human 

It must therefore be acknowledged, that that art which 
affords a cure to this weakness, or defect, of our under- 
standiiigs, and teaches us so to enumerate all the possible 
ways in which a given number of things may be mixed 
and combined together, that we may be certain that we 
have not omitted any one arrangement of them that can 


lead to the object of our inquiry, deserves to be considered 
as most emmently useful and worthy of our highest esteem 
and attention. And this is the business of the art or 
doctrine of combinations. Nor is this art or doctrine to be 
considered merely as a branch of the mathematical sciences. 
*or It has a relation to almost every species of useful know* 
ledge that the mmd of man can be employed upon It 
proceeds mdeed upon mathematical principles, in calculat- 
ing the number of the combinations of the things proposed • 
but by the conclusions that are obtained by it, the sagacity 
of the natural philosopher, the exactness of the historian, 
the skiU and judgment of the physician, and the prudence 
and foresight of the poUtician may be assisted; because 
the business of all these important professions is but to 
form reasonable conjectures concerning the several objects 
which engage their attention, and all wise conjectures are 
the results of a just and careful examination of the several 
different effects that may possibly arise from the causes 
tnat are capable of producing them." * 

Distinction of Combinaiims and Permutations. 

We must first consider the deep difference which exists 
between Combinations and Permutations, a difference in- 
volving important logical principles, and influencing the 
form of mathematical expressions. In permiUation we re- 
?Z'%r"'T''' ^^ order, treating AB as a different group 
irom liA. In combination we take notice only of the 
presence or absence of a certain thing, and pay no regard 
to Its place m order of time or space. Thus tie 1^ 
letters a, e, m, n can form but one combination, but thev 

We have hitherto been dealing with purely logical oiip«- 
tions, involving only combinattjn of qualUief I W 
fully pointed out in more than one placl that! thonV our 
symbols could not b.it be written in order of plkce aKad 
m orier of time, the relations expressed had^rregarf to 
pla^or tune (pp 33, r 14). TTie Law of Commutativeness 
in fact, expresses the condition that in logic we deal with 




combinations, and the same law is true of all the processes 
of algebra. In some cases, order may be a matter of 
indifference ; it makes no difference, for instance, whether 
gunpowder is a mixture of sulphur, carbon, and nitre, or 
carbon, nitre, and sulphur, or nitre, sulphur, and carbon, 
provided that the substances are present in proper propor- 
tions and well mixed. But this indifference of order does 
not usually extend to the events of physical science or the 
operations of art. The change of mechanical energy into 
heat is not exactly the same as the change from heat into 
mechanical energy ; thunder does not indifferently precede 
and follow lightning ; it is a matter of some importance 
that we load, cap, present, and fire a rifle in this precise 
order. Time is the condition of all our thoughts, space of 
all our actions, and therefore both in art and science we 
are to a great extent concerned with permutations. 
Language, for instance, treats different permutations of 
letters as having different meanings. 

Permutations of things are far more numerous than 
combinations of those things, for the obvious reason that 
each distinct thing is regarded differently according to 
its place. Tlius the letters A, B, C, will make different 
permutations according as A stands first, second, or third ; 
having decided the place of A, there are two places 
between which we may choose for B ; and then there 
remains but one place for C. Accordingly the permuta- 
tions of these letters will be altogether 3x2x1 or 6 in 
number. With four things or letters. A, B, C, D, wo 
shall have four choices of place for the first letter, three 
for the second, two for the third, and one for the fourth, 
80 that there will be altogether, 4x3x2x1, or 24 
permutations. The same simple rule applies in all cases ; 
beginning with the whole number of things we multiply 
at each step by a number decreased by a unit. In general 
language, if n be the number of things in a combination, 
the number of permutations is 

71 (n — i) (n — 2) 4.3.2. I. 

If we were to re-arrange the names of the days of 
the week, the possible arrangements out of which we 
should have to choose the new order, would be no less 
than 7 . 6 . 5 . 4 . 3 . 2 . I, or 5040, or, excluding the 
existing order, 5039. 




The reader will see that the numbers which we reach in 
questions of permutation, increase in a more extraordinary 
manner even than in combination. Each new object or 
term doubles the number of combinations, but increases 
the permutations by a factor continually growing. Instead 

of 2X2X2X2X we have 2X3X4X5X 

.and the products of the latter expression immensely 

exceed those of the former. These products of increasing 
factors are frequently employed, as we shall see, in ques- 
tions both of permutation and combination. They are 
technically called factorials, that is to say, the product of 
all integer numbers, from unity up to any number n is the 
factonal of n, and is often indicated symbolically by \n 
I give below the factorials up to that of twelve :— 

24 = I . 2 . 3 . 4 

120= I . 2 5 

720 = I . 2 6 

5,040 = [7 
40,320 = L8 

362,880 = L9 

3,628,800 = |ip 
39,916,800 - |ii 
479.001,600 = (12 
The factorials up to [36 are given in Rees's ' Cyclopedia, 
art. Cipher and the logarithms of factorials up to I265 

TJ^^.A ^'^'T^ ^J ^^^ ^^^ ^^ *^« ^ble of logarithms 
published under the superintendence of the Society for 
he Biffusion of Useful Knowledge (p. 215). To express 
the factorial I265 would require 529 places of figures 

Many wnters have from time to time remarked upon 
tlie extraordinary magnitude of the numbers with which 
we deal in this subject. Tacquet calculated ^ that the 
twenty.four letters of the alphabet may be arranged in 
more than 620 thousand trillions of ordera ; and Schott 
estimated that if a thousand millions of men were em- 
ployed for the same number of years in writing out these 
arrangements, and each man filled each day forty pages 
with forty arrangements in each, they would not Imve 
accomplished the task, as they would have written onl} 
584 thousand trillions instead of 620 thousand trillions. 

J Arithmetica Theoria. Ed. Amsteid. 1704. p C17 
• Rees's Cyclopadia, art Cipher. -^ ^ ^ f 

N 2 


f ■ 







In some questions the number of permutations may be 
restricted and reduced by various conditions. Some 
things in a group may be undistinguishable from others, 
so that change of order will produce no difference. Thus 
if we were to permutate the letters of the name Ann, 
according to our previous rule, we should obtain 3x2x1, 
or 6 orders ; but half of these arrangements would be 
identical with the other half, because the interchange of 
the two ns has no effect. The really different orders will 

"5 2 1 

therefore be ^ — '— or 3, namely Ann, Nan, Nna., In 

the word ntility there are two I's and two t'&, in respect 
of both of which pairs the numbers of permutations must 

be halved. Thus we obtain 

or 1260, as 

1 . 2 . 1 . 2 

the number of permutations. The simple rule evidently 
is — when some things or letters are undistinguished, 
proceed in the first place to calculate all the possible 
permutations as if all were different, and then divide by 
th6 numbers of possible permutations of those series of 
things which are not distinguished, and of which the 
permutations have therefore been counted in excess. 
Thus since the word Utilitarianisvi contains fourteen 
letters, of which four are i*s, two as, and two fs, the 
number of distinct arrangements will be found by 
dividing the factorial of 14, by the factorials of 4, 2, 
and 2, the result being 908,107,200. From the letters 
of the word Mississippi we can get in like manner 

, 1= j- or 34,650 permutations, which is not the one- 

[4 XLi X [2 

thousandth part of what we should obtain were all the 
letters different 

Calculation 0/ Number of Combinations. 

Although in many questions both of art and science 
we need to calculate the numbtir of permutations on 
account of their own interest, it far more frequently 
happens in scientific subjects that the)' possess but an 
indirect interest. As I have already pointed out, we 
almost always deal in the logical and mathematical 
sciences witli combinaiions, and varie^ of order enters 

only through the inherent imperfections of our symbols 
and modes of calculation. Signs must be used in some 
order, and we must withdraw our attention from this order 
before the signs correctly represent the relations of things 
which exist neither before nor after each other. Now, it 
often happens that we cannot choose all the combinations 
of things, without first choosing them subject to the 
accidental variety of order, and we must then divide by 
the number of possible variations of order, that we may 
get to the true number of pure combinations. 

Suppose that we wish to determine the number of ways 
in which we can select a group of three letters out of the 
alphabet, without allowing the same letter to be repeated. 
At the first choice we can take any one of 26 letters ; at 
the next step there remain 25 letters, any one of which 
may be joined with that already taken ; at the third step 
there will be 24 choices, so that apparently the whole 
number of ways of choosing is 26 x 25 x 24. But the 
fact that one choice succeeded another has caused us to 
obtain the same combinations of letters in different orders ; 
we should get, for instance, a, p, r at one time, and^, r, a at 
another, and every three distinct letters will appear six 
times over, because three things can be arranged in six 
permutations. To get the number of combinations, then, 
we must divide the whole number of ways of choosing, 
by six, the number of permutations of three things, 

26 X 25 X 24 


or 2,600. 

obtaining — ^ 

It is apparent that we need the doctrine of combina- 
tions in order that we may in many questions counteract 
the exaggerating effect of successive selection. If out of 
a senate of 30 persons we have to choose a committee of 5, 
we may choose any of 30 first, any of 29 next, and so on, 
in fact there will be 30 x 29 x 28 x 27 x 26 selections; 
but as the actual character of the members of the committee 
will not be affected by the accidental order of their selec- 
tion, we divide byi X2X3X4X5, and the possible 
number of different committees will be 142,506. Similarly 
if we want to calculate the number of ways in which the 
eight major planets may come into conjunction, it is evi- 
dent that they may meet either two at a time or three at 
a time, or four or more at a time, and as nothing is said iU 1*0 









f I 

the relative order or place in the conjunction, we require 
the number of combinations. Now a selection of 2 out of 8 

is possible in ;^l or 28 ways ; of 3 out of 8 in ?^ 

^•2 1.2.3 

or 56 ways ; of 4 out of 8 in ^'^'^'^ or 70 ways ; and it 

may be similarly shown that for 5, 6, 7, and 8 planets, 
meeting at one time, the numbers of ways are 56, 28, 8, 
and I. Thus we have solved the whole question of the' 
variety of conjunctions of eight planets ; and adding all the 
numbers together, we find that 247 is the utmost possible 
number of modes of meeting. 

In general algebraic language, we may say that a group 
of m things may be chosen out of a total number of n 
things, in a number of combinations denoted by the 

n . (n- I) (n~2) (n-3) (n-m + i ) 

''2 . 3.4.... n 
The extreme importance and significance of this formula 
seems to have been firet adequately recognised by Pascal, 
although its discovery is attributed by him to a friend, M.' 
de Gani^res.^ We shall find it perpetually recurring in 
questions l)oth of combinations and probability, and 
throughout the formulae of mathematical analysis traces 
of its influence may be noticed. 

Th€ AHthmeticcU Triangle, 

The Arithmetical Triangle is a name long since given to 
a series of remarkable numbers connected with the subject 
we are treating. According to Montucla « '* this triangle is 
in the theory of combinations and changes of order, almost 
what the table of Pythagoras is in ordinary arithmetic, that 
is to say, it places at once under the eyes the numbers re- 
quired in a multitude of cases of this theory." As early 
is 1544 Stifels had noticed the remarkable properties of 
these numbers and the mode of their evolution. Briggs, 
the inventor of the common system of logarithms, was *80 
struck with their importance that he called them the 

> (Buvres CompUtes de Pascal (1865), vol. iii. p. 302. Montucla 
states the name «s De Gruiires, Histoire des Mathematiqua, vol iii 
P- 389- 

' UUtoire des Mathefimtiquet, vol iii. p. 378. 

Abacus Panchrestus. Pascal, however, was the first who 
wrote a distinct treatise on these numbers, and gave them 
the name by which they are still known. But Pascal did 
not by any means exhaust the subject, and it remained for 
James Bernoulli to demonstrate fully the importance of 
the figurate numbers, as they are also called. In his 
treatise De Arte Conjectandi, he points out their applica- 
tion in the theoiy of combinations and probabilities, and 
remarks of the Arithmetical Triangle, " It not only con- 
tains the clue to the mysterious doctrine of combinations, 
but it is also the ground or foundation of most of the im- 
portant and abstruse discoveries that have been made in 
the other branches of the mathematics." ^ 

The numbers of the triangle can be calculated in a 
very easy manner by successive additions. We commence 
with unity at the apex ; in the next line we place a second 
unit to the right of this ; to obtain the third line of figures 
we move the previous line one place to the right, and add 
them to the same figures as they were before removal ; we 
can then repeat the same process ad infinitum. The 
fourth line of figures, for instance, contains i, 3, 3, i ; 
moving them one place and adding as directed we obtain : — 

Fourth line . . . 









Fifth line .... 











Sixth line .... 







Carrying out this simple process through ten more stepa 
we obtain the first seventeen lines of the Arithmetical 
Triangle as printed on the next page. Theoretically 
speaking the Triangle must be regarded as infinite in 
extent, but the numbers increase so rapidly that it soon 
becomes impracticable to continue the table. The longest 
table of the numbers which I have found is in Fortia's 
** IVait^ des Progressions " (p. 80), where they are given up 
to t)ie fortieth line and the ninth column. 

» Bernoulli, De Arte Ckmjeetandiy translated by Fi-ancis Maaerea. 
lioudon, I795i P* 75* 



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umm op vp tA) tovp 00 M to O 


^ ►* W to ^ »0>0 tM» 0« O M (i to ^ u>0 

•^Q|rj •" « to ^ mvp l>»QO Ov O •■ « to V w%>0 




Examining these numbers, we find that they are con- 
nected by an unlimited series of relations, a few of the 
more simple of which may be noticed. Each vertical 
column of numbers exactly corresponds with an oblique 
series descending from left to right, so that the triangle is 
perfectly symmetrical in its contents. The first column 
contains only units; the second column contains the 
natural numhers, I, 2, 3, &c. ; the third column contains 
a remarkable series of numbers, I, 3, 6, 10, 15, &c., which 
have long been called tJie triangular numbers, because they 
correspond with the numbers of balls which may be 
arranged in a triangular form, thus — 

o o 


o o 


o o 



o o 




The fourth column contains the pyramidal numbers, so 
called because they correspond to the numbers of equal 
balls which can be piled in regular triangular pyramids. 
Their differences are the triangular numbers. The numbers 
of the fifth column have the pyramidal numbers for their 
differences, but as there is no regular figure of which they 
express the contents, they have been arbitrarily called the 
trianguli-triangular numbers. The succeeding columns 
have, in a similar manner, been said to contain the 
trianguli-pyramickU, the pyramidi-pyramidal numbers, 
and so on.^ 

From the mode of formation of the table, it follows that 
the differences of the numbers in each column will be 
found in the preceding column to the left. Hence the 
second differenceSy or the differences of differences^ will be 
in the second column to the left of any given column, the 
third differences in the third column, and so on. Thus 
we may say that unity which appears in the first column 
is the first difference of the numbers in the second column ; 
the second difference of those in the third column ; the third 
difftrence of those in the fourth, and so on. The triangle 
is seen to be a complete classification of all numbers 
according as they have unity for any of their differences. 

Since each line is formed by adding the previous line 

( Willis's Algtbra, Discourse of Combinations, ^q., p. 109. 







to itself, it is evident that the sum of the numbers in each 
horizontal line must be double the sum of the numbers in 
the line next above. Hence we know, without making 
the additions, that the successive sums must be i, 2, 4, 
8, 16, 32, 64, &c., the same as the numbers of combinations 
in the Logical Alphabet. Speaking generally, the sum of 
the numbers in the nth line will be 2*-\ 

Again, if the whole of the numbers down to any line be 
added together, we shall obtain a number less by unity 
than some power of 2; thus, the first line gives i or 
2^— I ; the first two lines give 3 or 2^— 1 ; the finst three 
lines 7 or 2^ — i ; the first six lines give 63 or 2* — i ; or, 
speaking in general language, the sum of the first n lines 
is 2* — I. It follows that the sum of the numbers in any 
one line is equal to the sum of those in all the preceding 
lines increased by a unit. For the sum of the nth line is, 
as ah-eady shown, 2'^\ and the sum of the first ti— i lines 
is 2*^' — I, or less by a unit. 

This account of the properties of the figurate numbers 
does not approach completeness ; a considerable, probably 
an unlimited, number of less simple and obvious relations 
might be traced out. Pascal, after giving many of the 
properties, exclaims ^ : "Mais j'en laisse bien plus que je 
n'en donne ; c'est une chose Strange combien il est fertile 
en propri^tes! Chacun pent s'y exercer." The arith- 
metical triangle may be considered a natural classification 
of numbers, exhibiting, in the most complete manner, 
their evolution and relations in a certain point of view. 
It is obvious that in an unlimited extension of the 
triangle, each number, with the single exception of the 
number two, has at least two places. 

Though the properties above explained are highly 

curious, the greatest value of the triangle arises from the 

fact that it contains a complete statement of the values of 

the formula (p. 182), for the numbers of combinations of m 

things out of n, for all possible values of ni and n. Out 

of seven things one may be chosen in seven ways, and 

seven occurs in the eighth line of the second column. The 

combinations of two things chosen out of seven are 


or 21, which is the third number in the eighth 

I X 2 

* UHuvr^ CompUks^ vol, iii. p, 251. 

■'f I J n> «■ 



line. The combinations of three things out of seven are 


1X2X3 ^^ ^5' which appeai-s fourth in the eighth line. 

In a similar manner, in the fifth, sixth, seventh, and eighth 
columns of the eighth line I find it stated in how many 
ways I can select combinations of 4, 5, 6, and 7 things out 
of 7. Proceeding to the ninth line, I find in succession 
the number of ways in which I can select i, 2, 3, 4, 5, 6 
7, and 8 things, out of 8 things. In general lan^age, if 
I wish to know in how many ways m things can be 
selected in combinations out of n things, I must look in 
the n + I**' line, and take the wi + i** number, as the 
answer. In how many ways, for instance, can a sub- 
committee of five be chosen out of a committee of nine. 
The answer is 126, and is the sixth number in the tenth 
line; it will be found equal to 9 ■ « » 7 . 6^5^ ^^^^y^ 
our formula (p. 182) gives. * -3.4.5 

The full utility of the figurate numbers will be more 
apparent when we reach the subject of probabilities, but I 
may give an illustration or two in this place. In how 
many ways can we arrange four pennies as regards head 
and taU ? The question amounts to asking in how many 
ways we can select o, i, 2, 3, or 4 heads, out of 4 heads, 
and the fifth line of the triangle gives us the complete 
answer, thus — 

We can select No head and 4 tails in i way. 
„ I head and 3 tails in 4 ways. 

M 2 heads and 2 tails in 6 ways. 

>, 3 heads and i tail in 4 ways. 

„ 4 heads and o tail in i way. 

The total number of different cases is 16, or 2* and 
when we come to the next chapter, it will be found that 
these nuinbers give us the respective probabilities of all 
throws with four pennies. 

I ^ve in p. 181 a calculation of the number of ways in 
which eight planets can meet in conjunction ; the reader 
will find all the numbers detailed in the ninth line of the 
anthmetical triangle. The sum of the whole line is 2» or 
256; but we must subtract a unit for the case where no 
planet appears, and 8 for the 8 cases in which only one 
planet appears; so that the total number of conjunctions 





is 2* — I — 8 or 247. If an organ has eleven stops we 
find in the twelfth line the numbers of ways in which we 
can draw them, i, 2, 3, or more at a time. Thus there are 
462 ways of drawing five stops at once, and as many of 
drawing six stops. The total number of ways of varying 
the sound is 2048, including the single case in which no 
stop at all is drawn. 

One of the most important scientific uses of the arith- 
metical triangle consists in the information which it gives 
concerning the comparative frequency of diveigencies 
from an average. Suppose, for the sake of argument, that 
all persons were naturally of the equal stature of five feet, 
but enjoyed during youth seven independent chances of 
growing one inch in addition. Of these seven chances, 
one, two, three, or more, may happen favourably to any 
individual; but, as it does not matter what the chances 
are, so that the inch is gained, the question really turns 
upon the number of combinations of o, I, 2, 3, &c., things 
out of seven. Hence the eighth line of the triangle gives 
us a complete answer to the question, as follows : — 

Out of every 128 people — 


One person would have the stature of 

7 persons 
21 persons 
35 persons 
35 persons 
21 persons 

7 persons 

I person 














By taking a proper line of the triangle, an answer may 
be had under any more natural supposition. This theory 
of comparative frequency of divergence fi-om an average, 
was first adequately noticed by Quetelet, and has lately 
been employed in a very interesting and bold manner 
by Mr. Francis Galton,^ in his remarkable work on 
" Hereditary Genius." We shall afterwards find that the 
theory of error, to which is made the ultimate appeal in 
cases of quantitative investigation, is founded upon the 

* See also Galton's Lecture at the Royal Institution, 27th February, 
1874 ; Catalogue of the Spocial Loan Collection of Scientific Instru- 
ments, South Kensington, Nob. 48 49 ; and Galton, FhiiosophictU 
Ma^iuine, January 1875, 



comparative numbers of combinations as displayed in the 

Connection between the Arithmetical TrtOngle and the 

Logical Alphabet, 

There exists a close connection between the arithmetical 

triangle described in the last section, and the series of 

combinations of letters called the Logical Alphabet. The 

one is to mathematical science what the other is to 

logical science. In fact the figurate numbers, or those 

exhibited in the triangle, are obtained by summing up the 

logical combinations. Accordingly, just as the total of the 

numbers in each line of the triangle is twice as great as 

that for the preceding line (p. 186), so each column of the 

Alphabet (p. 94) contains twice as many combinations as 

the preceding one. The like correspondence also exists 

between the sums of all the lines of figures down to any 

particular line, and of the combinations down to any 

particular column. 

By examining any column of the Logical Alphabet we 
find that the combinations naturally group themselves 
according to the figurate numbers. Take the combinations 
of the letters A, B, C, D ; they consist of all the ways in 
which I can choose four, three, two, one, or none of the 
four letters, filling up the vacant spaces with negative 


There is one combination, ABCD, in which all the 
positive letters are present ; there are four combinations in 
each of which three positive letters are present; six in 
which two are present ; four in which only one is present ; 
and, finally, there is the single case, abed, in which all 
positive letters are absent. These numbers, i, 4, 6, 4, i, 
are those of the fifth line of the arithmetical triangle, and 
a like correspondence will be found to exist in each 
column of the Ix)gical Alphabet. 

Numerical abstraction, it has been asserted, consists in 
overlooking the kind of difference, and retaining only a 
consciousness of its existence (p. 158). While in logic, 
then, we have to deal with each combination as a separate 
kind of thing, in arithmetic we distinguish only the classes 
which depend upon more or less positive terms being 





rif' < 


present, and the numbers of these classes immediately 
produce the numbers of the arithmetical triangle 

It may here be pointed out that there are two modes in 
which we can^alculate the whole number of combinations 
01 certam thmgs. Either we may take the whole number 
at once as shown in the Ixjgical Alphabet, in which case 
the number will be some power of two, or else we may 
calculate successively, by aid of permutations, the number 
ot combinations of none, one, two, three things, and so 
on. Hence we arrive at a necessary identity between two 
have' °""'^'*^- '" ^^'"^ ^^e «^ ^^^^^ <'hings we shall 

2 = I 4- 1 + 4_^ 3 , 4 > 3 > 2 4 3. 2 ^i 
I '•2^i.2.3"^"i.2.34- 
In a general form of expression we shall have 

' 1.2 ^ 1.2.3 • ' 

the terms being continued until they cease to have any 
value. Thus we arrive at a proof of simple cases of the 
Binomial Theorem, of which each column of the Logical 
Alphabet IS an exemplification. It may be shown that aU 
other mathematical expansions likewise arise out of simple 
processes of combination, but the more complete considera- 
tion of this subject must be deferred to another work. 

Possible Variety of Nature and Art 

We cannot adequately understand the difficulties 
which beset us in certain branches of science, unless we 
have some clear idea of the vast numbers of combinations 
or permutations which may be possible under certain con- 
ditions Thus only can we learn how hopeless it would 
be to attempt to treat nature in detail, and exhaust the 
whole number of events which might arise. It is instruc- 
tive to consider, in the first place, how immensely great 
are the numbers of combinations with which we deal in 
many arts and amusements. 

In dealing a pack of cards, the number of hands, of 
thirteen cards each, which can be produced is evidently 
52 X 51 X 50 X . . . X 40 divided by i x 2 x ^ x lit 
or 635,013,559,600. But in whist four hands are' simut' 




taneously held, and the number of distinct deals becomes 
so vast that it would require twenty-eight figures to express 
it If the whole population of the world, say one thousand 
millions of persons, were to deal cards day and night, for 
a hundred million of years, they would not in that time 
have exhausted one hundred-thousandth part of the pos- 
sible deals. Even with the same hands of cards the play 
may be almost infinitely varied, so that the complete 
variety of games at whist which may exist is almost 
incalculably great. It is in the highest degree improbable 
that any one game of whist was ever exactly like another, 
except it were intentionally so. 

The end of novelty in art might well be dreaded, did 
we not find that nature at least has placed no attainable 
limit, and that the deficiency will lie in our inventive 
faculties. It would be a cheerless time indeed when all 
possible varieties of melody were exhausted, but it is 
readily shown that if a poal of twenty-four bells had been 
rung continuously from the so-called beginning of the 
world to the present day, no approach could have been 
made to the completion of the possible changes. Nay, 
had every single minute been prolonged to 10,000 years, 
still the task would have been unaccomplished.^ As 
regards ordinary melodies, the eight notes of a single 
octave give more than 40,000 permutations, and two 
octaves more than a million millions. If we were to take 
into account the semitones, it would become apparent that 
it is impossible to exhaust the variety of music. When 
the late Mr. J. S Mill, in a depressed state of mind, feared 
the approaching exhaustion of musical melodies, he had 
certainly not bestowed sufficient study on the subject of 

Similar considerations apply to the possible number of 
natural substances, though we cannot always give precise 
numerical results. It was recommended by Hatehett ' 
that a systematic examination of all alloys of metals 
should be carried out, proceeding from the binary ones to 
more complicated ternary or quaternary ones. He can 
hardly have been aware of the extent of his proposed 


* Wallis, Of Combinations^ p. 116, quoting Vossius. 

* Philosophical Transactiont (1803). vol. xciii. p. 193. 








inquiry. If we operate only upon thirty of the known 
metals, the number of binary alloys would be 435, of 
ternary alloys 4060, of quaternary 27,405, without paying 
regard to the varying proportions of the metals, and only 
regarding the kind of metal. If we varied all the ternary 
alloys by quantities not less than one per cent., the 
number of these alloys would be 11445,060. An ex- 
haustive investigation of the subject is therefore out of 
the question, and unless some laws connecting the proper- 
ties of the alloy and its components can be discovered, it 
is not apparent how our knowledge of them can ever be 
more than fragmentary. 

The possible variety of definite chemical compounds, 
again, is enormously great. Chemists have already ex- 
amined many thousands of inorganic substances, and a 
still greater number of organic compounds ; ^ they have 
nevertheless made no appreciable impression on the 
number which may exist. Taking the number of ele- 
ments at sixty-one, the number of compounds contain- 
ing different selections of four elements each would 
be more than half a million (521,855). As the same 
elements often combine in many different proportions, 
and some of them, especially carbon, have the power of 
forming an almost endless number of compounds, it 
would hardly be possible to assign any limit to the 
number of chemical compounds which may be formed. 
There are branches of physical science, therefore, of which 
it is unlikely that scientific men, with all their industry, 
can ever obtain a knowledge in any appreciable degree 
approaching to completeness. 

Higher Orders of Variety. 

The consideration of the facts already given in this 
chapter will not produce an adequate notion of the pos- 
sible variety of existence, unless we consider the com- 
parative numbers of combinations of different orders. By 
a combination of a higher order, I mean a combination 
of groups, which are themselves groups. The immense 
numbers of compounds of carbon, hydrogen, and oxygen, 

^ Uohuann'a Introduction to ChemxHry^ p. 36. 


described in organic chemistry, are combinations of a 
second order, for the atoms are groups of groups. The 
wave of sound produced by a musical instrument may be 
regarded as a combination of motions ; the body of sound 
proceeding from a large orchestra is therefore a complex 
aggregate of sounds, each in itself a complex combination 
of movements. All literature may be said to be developed 
out of the difference of white paper and black ink. From 
the unlimited number of marks which might be cliosen we 
select twenty-six conventional letters. The pronounceable 
combinations of letters are probably some trillions in 
number. Now, as a sentence is a selection of words, the 
possible sentences must be inconceivably more numerous 
than the words of which it may be composed. A book is 
a combination of sentences, and a library is a combination 
of books. A library, therefore, may be regarded as a com- 
bination of the fifth order, and the powers of numerical 
expression would be severely tasked in attempting to 
express the number of distinct libraries which might be 
constructed. The calculation, of course, would not be 
possible, because the union of letters in words, of words 
in sentences, and of sentences in books, is governed by 
conditions so complex as to defy analysis. I wish only to 
point out that the infinite variety of literature, existing or 
possible, is all developed out of one fundamental differ- 
ence. Galileo remarked that all truth is contained in the 
compass of the alphabet. He ought to have said that it 
is all contained in the difference of ink and paper. 

One consequence of successive combination is that the 
simplest marks will suffice to express any information. 
Francis Bacon proposed for secret writing a biliteral 
cipher, which resolves all letters of the alphabet into 
permutations of the two letttirs a and b. Thus A was 
aaaaa, B acuiab, X habab, and so on.i In a similar way, 
as Bacon clearly saw, any one difference can be made the 
ground of a code of signals ; we can express, aa he says, 
omnia per omnia. The Morse alphabet uses only a 
Bucceasion of long and short marks, and other systems 
of telegraphic language employ right and left strokes 
A single lamp obscured at various intervals, long or 

» Works, ediiod by Shaw, vol i pp. 141—145, quoted in Bees' 
EneyclopcBdia, art Cipher. 







short, may be made to spell out any words, and with 
two lamps, distinguished by colour, position, or any 
other circumstance, we could at once represent Bacon's 
biliteral alphabet. Babbage ingeniously suggested that 
every lighthouse in the world should be made to spell 
out its own name or number perpetually, by flashes or 
obscurations of various duration and succession. A 
system like that of Babbage is now being applied to 
lighthouses in the United Kingdom by Sir W. Thomson 
and Dr. John Hopkinson. 

Let us calculate the numbers of combinations of dif- 
ferent orders which may arise out of the presence or 
absence of a single mark, say A. In these figures 

| A|A | lAI I I !AJ l_J I 

we have four distinct varieties. Form them into a group 
of a higher order, and consider in how many ways we 
may vary that group by omitting one or more of the 
component parts. Now, as there are four parts, and any 
one may be present or absent, the possible varieties will 
be2X2X2X2, or i6in number. Form these into a new 
whole, and proceed again to create variety by omitting 
any one or more of the sixteen. The number of pos- 
sible changes will now be, or 
2", and we can repeat the process again and again. We 
are imj^ining the creation of objects, whose numbers are 
represented by the successive orders of the powers of two. 
At the fii-st step we have 2 ; at the next 2*, or 4 ; 

at the third 2^ , or 16, numbers of very moderate amount. 


Let the reader calculate the next term, 2* , and he will be 

surprised to find it leap up to 65,536. But at the next 

step he has to calculate the value of 65,536 tu'o*3 multiplied 

together, and it is so great that we could not possibly 

compute it, the mere expression of the result requiring 

19,729 places of figures. But go one step more and we 

pass the bounds of all reason. The sixth order of the 

powers of ttuo becomes so great, that we could not even 

express the number of figures required in writing it down, 

without using about 19,729 figures for the purpose. The 

successive orders of the powers of two have then the 

II >■ 




following values so far as we can succeed in describing 
them : — 

First order .... 2 

Second order . . . . 4 

Third order .... 16 

Fourth order .... 65,536 
Fifth order, number expressed by 19,729 figures. 
Sixth order, number expressed by 
figures, to express the number 
of which figures would require 
about .... 19,729 figures. 

It may give us some notion of infinity to remember 
that at this sixth step, having long surpassed all bounds 
of intuitive conception, we make no approach to a limit. 
Nay, were we to make a hundred such steps, we should be 
as far away as ever from actual infinity. 

It is well worth observing that our powers of expression 
rapidly overcome the possible multitude of finite objects 
which may exist in any assignable space. Archimedes 
showed long ago, in one of the most remarkable writings 
of antiquity, the Liber de Arencc Numero, that the grains of 
sand in the world could be numbered, or rather, that if 
numbered, the result could readily be expressed in arith- 
metical notation. Let us extend his problem, and ascertain 
whether we could express the number of atoms which could 
exist in the visible universe. The most distant stars which 
can now be seen by telescopes— those of the sixteenth 
magnitude— are supposed to have a distance of about 
33*900,000,000,000,000 miles. Sir W. Thomson has 
shown reasons for supposing that there do not exist 
more than from 3 x lo^* to lo^* molecules in a cubic 
centimetre of a solid or liquid substance.* Assuming 
these data to be true, for the sake of argument, a simple 
calculation enables us to show that the abnost inconceivably 
vast sphere of our stellar system if entirely filled with 
solid matter, would not contain more than about 6S x 10^ 
atoms, that is to say, a number requiring for its expression 
92 places of figures. Now, this number would be im- 
mensely less than the fifth order of the powers of two. 
In the variety of logical relations, which may exist 

» Nature, vo'. i P. 553 

o 2 




between a certain number of logical terms, we also meet 
a case of higher combinations. We have seen (p. 142) that 
with only six terms the number of possible selections of 
combinations is 18446,744,073,709,551,616. Considering 
that it is the most common thing in the world to use an 
argument involving six objects or terms, it may excite 
some surprise that the complete investigation of the 
relations in which six such terms may stand to each 
other, shoidd involve an almost inconceivable number 
of cases. Yet these numbers of possible logical relations 
belonc[ only to the second order of combinations. 

;;• ^ 

r i 



The subject upon which we now enter must not be 
reg:arded as an isolated and curious branch of speculation. 
It is the necessary basis of the judgments we make in the 
prosecution of science, or the decisions we come to in the 
conduct of ordinary affairs. As Butler truly said, " Pro- 
bability is the very guide of life." Had the science of 
numbers been studied for no other purpose, it must have 
been developed for the calculation of probabilities. All 
our inferences concerning the future are merely probable, 
and a due appreciation of the degree of probability depends 
upon a comprehension of the principles of the subject. I 
am convinced that it is impossible to expound the methods 
of induction in a sound manner, without resting them upon 
the theory of probability. Perfect knowledge alone can 
give certainty, and in nature perfect knowledge would be 
infinite knowledge, which is clearly beyond our capacities. 
We have, therefore, to content ourselves with partial 
knowledge — knowledge mingled with ignorance, producing 

A great difficulty in this subject consists in acquiring a 
precise notion of the matter treated. What is it that we 
number, and measure, and calculate in the theory of pro- 
babilities ? Is it belief, or opinion, or doubt, or knowledge, 
or chance, or necessity, or want of art ? Does probability 
exist in the things which are probable, or in the mind which 
regards them as such ? The etymology of the name lends 
us no assistance : for, curiously enough, probable is ultimately 
the same word as provable, a good instance of one word 
becoming differentiated to two opposite meanings. 



f \ 






! I 






Chance cannot be the subject of the theory, because 
there is really no such thing as chance, regarded as pro- 
ducing and governing events. The word cliance signifies 
falling, and the notion of falling is continually used as a 
simile to express uncertainty, because we can seldom pre- 
dict how a die, a coin, or a leaf will fall, or when a bullet 
will hit the mark. But everyone sees, after a little 
reflection, that it is in our knowledge the deficiency lies, 
not in the certainty of nature's laws. There is no doubt in 
lightning as to the point it shall strike; in the greatest 
storm there is nothing capricious ; not a grain of sand lies 
upon the beach, but infinite knowledge would account for 
its lying there ; and the course of every falling leaf is 
guided by the principles of mechanics which rule the 
motions of the heavenly bodies. 

Chance then exists not in nature, and cannot coexist 
with knowledge; it is merely an expression, as Laplace 
remarked, for our ignorance of the causes in action, and 
our consequent inability to predict the result, or to bring 
it about infallibly. In nature the happening of an event 
has been pre-determinod from the first fashioning of the 
universe. Prohahility belongs wholly to the mind. This is 
proved by the fact that different minds may regard the 
very same event at the same time with widely different 
degrees of probability. A steam-vessel, for instance, is 
missing and some persons believe that she has sunk in 
mid -ocean; others think differently. In the event itself 
there can be no such uncertainty ; the steam-vessel either 
has sunk or has not sunk, and no subsequent discussion of 
the probable nature of the event can alter the fact. Yet 
the probability of the event will really vary from day to 
day, and from mind to mind, according as the slightest 
information is gained regarding the vessels met at sea, the 
weather prevailing there, the signs of wreck picked up, 
or the previous condition of the vessel. Probability thus 
belongs to our mental condition, to the light in which we 
regard events, the occurrence or non-occurrence of which 
is certain in themselves. Many writers accordingly have 
asserted that probability is concerned with degree or 
quantity of belief. De Morgan says,^ " By degree of proba- 

* Formal Lo^^ p. 172. 





bility we really mean or ought to mean degree of belief." 
The late Professor Don kin expressed the meaning of 
probability as " quantity of belief ; " but I have never felt 
satisfied with such definitions of probability. The nature 
of belief is not more clear to my mind than the notion 
which it is used to define. But an all-sufficient objection 
is, that the theory does not measure what the belief is, but 
what it ought to be. Few minds think in close accordance 
with the theory, and there are many cases of evidence in 
which the belief existing is habitually different from what 
it ought to be. Even if the state of belief in any mind 
could be measured and expressed in figures, the results 
would be worthless. The value of the theory consists in 
coiTecting and guiding our belief, and rendering our states 
of mind and consequent actions harmonious with our 
knowledge of exterior conditions. 

This objection has been clearly perceived by some of 
those who still used quantity of belief as a definition of 
probability. Thus De Morgan adds — "Belief is but 
another name for imperfect knowledge." Donkin has 
well said that the quantity of belief is " always relative 
to a particular state of knowledge or ignorance; but it 
must be observed that it is absolute in the sense of not 
being relative to any individual mind; since, the same 
information being presupposed, all minds ought to dis- 
tribute their belief in the same way." ^ Boole seemed to 
entertain a like view, when he described the theory as 
engaged with "the equal distribution of ignorance;"* 
but we may just as well say that it is engaged with the 
equal distribution of knowledge. 

I prefer to dispense altogether with this obscure word 
belief, and to say that the theory of probability deals with 
qtiantiiy of knowledge, an expression of which a precise 
explanation and measure can presently be given. An 
event is only probable when our knowledge of it is 
diluted with ignorance, and exact calculation is needed 
to discriminate how much we do and do not know. The 
theory has been described by some writers as professing to 
evolve knowledge out of ignorance ; but as Donkin admirably 
remarked, it is really " a method of avoiding the erection 

' Philosophical Afagazine, 4th Series, vol. i. p. 355. 

* Transa4;tumt of the Royal Society of Edinburghy vol. x»i. paxt 4, 


' i 








hv ^^}f 'i.P"" ^gnorance." It defines rational expectation 
by measuring he comparative amounts of knowledge and 
Jgno^nce and teaches us to regulate our Jionf w*tS 
regard to future events in a way which wUl. in the lone 
run. lead to the least disappointment It is, a.s llpl^ 
happUy said, ffood stnseredn^ to calculation. This theorv 
appears to me the noblest creation of intellect andH 
passes ^y conception how two such men as Auguste Oomte 
and J. S. Mill could be found depreciating it and vainlv 
questioning its validity. To eulojise the theory ou^h to 
be as needless as to eulogise reason itself. 

Fundamental Principles of the Thevry. 

The calculation of probabilities is really founded as I 
conceive upon the principle of reasoning set forth in pre 

we kLwTr'- ^' "•"'' "-^"^ «*1'"J^ equally and7h^ 
we know of one case may be affirmed of every case 
reserablmg ,t m the necessaiy circumstances. The theo^ 
consists in putting similar ca-ses on a par, and distribS 
equally among them whatever knowled^ wf ^sS 
Throw a penny into the air, and consider^whlt wHuow 
with regard to its way of falling. We know that it 3 
certainly fall upon a side, so that either head or tail w 
be uppermost; but as to whether it will be head or Ta 
our knowledge IS equally divided. Whatever we know 
concerning head, we know also concerning tail, so that we 

ThlwV'-T" ^"^ ^xP^cting one more "than the othlr 
The least predominance of belief to either side would bo 
irrational; it would consist in treating unequal^ thinS 
of which our knowledge is equal. ^ ^ 

The theory does not require, as some writers have 
ei-roneously supposed, that we should first ^Zi^Z 

sXr^'c^''? ^-l'"^ ^'^^'y of *»>« events ^Z\Z 
sidcnng. So far as we can examine and measure the 

oTpXbihr'Th'eT'^ "^ "'""^^ ""' of ZTph'ere 
01 proDabUity. The theory comes into play where ionor 

ance begins, and the knowledge we possJL requfres tote 
distributed over many cases. Nor docs the thTr^ show 
that the com will fall as often on the one side TZ otheT 
Tt IS almost impossible that this should hap^n because 
»ome inequality in the form of the coin, or S u^form 





manner in throwing it up, is almost sure to occasion a 
slight preponderance in one direction. But as we do not 
previously know in which way a preponderance will exist 
we have no reason for expecting head more than tail Our 
state of knowledge wiU be changed should we throw up 
the com many times and register the results. Every throw 
gives us some slight information as to the probable 
tendency of the coin, and in subsequent calculations we 
must take this into account. In other cases experience 
might show that we had been entirely mistaken ; we mi^ht 
expect that a die would faU as often on each of the lix 
sides as on each other side in the long run ; trial might show 
that the die was a loaded one, and falls most often on a 
particular face. The theory would not have misled us • it 
treated correctly the information we had, which is all that 
any theory can do. 

It may be asked, as Mill asks, Why spend so much 
trouble m calculating from imperfect data, when a little 
trouble would enable us to render a conclusion certain bv 
actual trial ? Why calculate the probabUity of a measure- 
ment being correct, when we can try whether it is correct ? 
But I shall fully point out in later parts of this work that 
m measurement we never can attain perfect coincidence 
Two measurements of the same base line in a survey may 
show a difference of some inches, and there may be no 
means of knowing wliich is the better result. A third 
measurement would probably agree with neither. To 
select any one of the measurements, would imply that 
we knew it to be the most nearly coiTect one, which we 
do not In this state of ignorance, the only guide is the 
theory of probabihty, which proves that in the lon<T run 
the mean of divergent results wUl come most neariy to 
the truth. In aU other scientific operations whatsoever 
perfect knowledge is impossible, and when we have ex- 
hausted all our instrumental means in the attainment of 
truth, there is a margin of error which can only be safely 
treated by the principles of probability. 

The method which we employ in the theory consists in 
calculating the number of all the cases or events concerning, 
which our knowledge is equal. If we have the slightest 
reason for suspecting that one event is more likely to 
occur than another, we should take this knowledge into 





r ., i 





accouni This being done, we must determine the whole 
number of events which are, so far as we know, equally 
likely. Thus, if we have no reason for supposing that a 
penny will fall more often one way than another, there are 
two cases, head and tail, equally likely. But if from trial 
or otherwise we know, or think we know, that of loo 
throws 55 will give tail, then the probability is measured 
by the ratio of 55 to loa 

The mathematical formulae of the theory are exactly the 
same as those of the theory of combinations. In this 
latter theory we determine in how many ways events may 
be joined together, and we now proceed to use this know- 
ledge m calculating the number of ways in which a certain 
event may come about. It is the comparative numbers of 
ways m which events can happen which measure their 
comparative probabQities. If we throw three pennies 
into the air, what is the probability that two of them 
will fall taa uppermost ? This amounts to askin.^ in how 
many possible ways can we select two tails out%f three 
compared with the whole number of ways in which the 
coins can be placed. Now, the fourth line of the Arith- 
metical Triangle (p. 184) gives us the answer. The whole 
number of ways in which we can select or leave three thinrrs 
IS eight, and the possible combinations of two things at^'a 
time IS three ; hence the probability of two tails^'is the 
ratio of three to eight. From the numbers in the trian<Tle 
we may similarly draw all the following probabilities :-^ 
One combination gives o tail. Probability J. 
Three combinations gives i tail Probability f . 
Three combinations give 2 tails. Probability f . 
One combination gives 3 tails. Probability J. 
We can apply the same considerations to the imaginary 
causes of the difference of stature, the combinations of 
which were shown in p. 188. There are altogether 128 
ways m which seven causes can be present or absent. 
Now, twenty-one of these combinations give an addition 
of two inches, so tliat the probability of a person under 
the circumstances being five feet two inches is ^. The 
probability of five feet three inches is yVff ; of five feet 
one inch ^ ; of five feet ^4^, and so on. Thus the 
eighth line of the Arithmetical Triangle gives all the 
probabilities arising out of the combinations of seven causes. 




Bules for tlie CalctclcUion of Probabilities, 

I will now explain as simply as possible the rules 
for calculating probabilities. The principal rule is as 
follows : — 

Calculate the number of events which may happen 
independently of each other, and which, as far as is 
known, are equally probable. Make this number the 
denominator of a fraction, and take for the numerator 
the number of such events as imply or constitute tne 
happening of the event, whose probability is required. 

Thus, if the letters of the word Roma be thrown down 
casually in a row, what is the probability that they will 
form a significant Latin word ? The possible arrange- 
ments of four letters are 4 X 3 x 2 x i, or 24 in number 
(p. 178), and if all the arrangements be examined, seven 
of these will be found to have meaning, namely Roma, 
ramo, oram, mora, maro, arm^, and amor. Hence the 
probability of a significant result is ^, 

We must distinguish comparative from absolute pro- 
babilities. In drawing a card casually from a pack, there 
is no reason to expect any one card more than any other. 
Now, there are four kings and four queens in a pack, so 
that there are just as many ways of drawing one as the 
other, and the probabilities are equal. But there are 
thirteen diamonds, so that the probability of a king is to 
that of a diamond as four to thirteen. Thus the probabili- 
ties of each are proportional to their respective numbers 
of ways of happening. Again, I can draw a king in four 
ways, and not draw one in forty-eight, so that the pro- 
babilities are in this proportion, or, as is commonly said, 
the odiis against drawing a king aro forty-eight to four. 
The odds are seven to seventeen in favour, or seventeen to 
seven against the letters R,o,m,a, accidentally forming a 
significant word. The odds are five to three against two 
tails appearing in three throws of a penny. Conversely, 
when the odds of an event are given, and the probability is 
required, take the odds in favour of the event for numerator, 
and the sum of the odds for denominator. 

It is obvious that an event is certain when all the com- 
binations of causes which can take place produce that 
event If we represent the probability of such event 

' i 





according to our rule, it gives the ratio of some number to 
itself, or unity. An event is certain not to happen when 
no possible combination of causes gives the event, and the 
ratio by the same rule becomes tliat of o to some' number 
Hence it follows that in the theory of probability certainty 
IS expressed by i, and impossibility by o ; but no mystical 
meaning should be attached to these symbols, as they 
merely express the fact that all or tw possible combinations 
give the event. 

By a compound event, we mean an event which may be 
decomposed into two or more simpler events. Thus the 
firing of a gun may be decomposed into puUin^ the 
trigger, the fall of the hammer, the explosion of the 
cap, &C. In this example the simple events are not 
independcTU, because if the trigger is pulled, the other 
events will under proper conditions necessarily follow, and 
their probabilities are therefore the same as that of the 
first event Events are independent when the happenin<T 
of one does not render the other either more or le^ 
probable than before. Thus the death of a person is 
neither more nor less probable because the planet Mars 
happens to be visible. When the component evente are 
independent, a simple rule can be given for calculatincr 
the probabihty of the compound eveut, thuH^Multiply 
together the fractions expressing the probabilities of the 
independent component events. 

The probability of throwing tail twice with a penny is 
* X i, or i ; the probabUity of throwing it three times 
running is i x J x J, or J ; a result agreeing with that 
obtained m an apparently diiierent manner (p. 202). In 
fact, when we multiply together the denominators, we 
get the whole number of ways of happening of the com- 
pound event, and when we multiply the numerators, we 
get the number of ways favourable to the required event 

Probabilities may be added to or subtracted from each 
other under the important condition that the events in 
question are exclusive of each other, so that not more than 
one of them can happen. It might be argued that, since 
the probability of throwing head at the first trial is i, and 
at the second tnal also i, the probability of throwing it 
in the firat two throws is ^ + J, or certainty. Not only is 
this result evidentlv absurd, but a repetition of the process 




would lead us to a probability of i^ or of any greater 
number, results which could liave no meaning whatever. 
The probability we wish to calculate is that of one head in 
two throws, but in our addition we have included the case 
in which two heads appear. The tme result is J + J x ^ 
or }, or the probability of head at the first throw, added to 
the exclusive probability that if it does not come at the 
first, it will come at the second. The greatest difficulties 
of the theory arise from the confusion of exclusive and 
unexclusive alternatives. I may remind the reader that 
the possibility of unexclusive alternatives was a point 
previously discussed (p. 68), and to the reasons then given 
for considering alternation as logically unexclusive, may 
be added the existence of these difficulties in the theory of 
probability. The erroneous result explained above really 
arose from overlooking the fact that the expression " head 
first throw or head second throw " might include the case 
of head at both throws. 

The Logical Alphabet in questions of Probability. 

When the probabilities of certain simple events are 
given, and it is required to deduce the probabilities of 
compound events, the Logical Alphabet may give assist- 
ance, provided that there are no special logical conditions 
so that all the combinations are possible. Thus, if there be 
three events, A, B, C, of which the probabilities are, a, ^, 
7, then the negatives of those events, expressing the absence 
of the events, will have the probabilities i — a, i —fi, 1—7. 
We have only to insert these values for the letters of the 
combinations and multiply, and we obtain the probability 
of each combination. Thus the probability of ABC is 
aJ3y; of Abe, a(l - /9)(l - 7). 

We can now clearly distinguish between the probabilities 
of exclusive and unexclusive events. Thus, if A and B 
are events which may happen together like rain and high 
tide, or an earthquake and a stoi-m, the probability of A or 
B happening is not the sum of their separate probabilities. 
For by the Laws of Thought we develop A -I- B into 
AB'|'A5«|»aB, and substituting a and 0, the probabili- 
ties of A and B respectively, we obtain a.l3-\-a.(l — /9)-h 
(l— a).)9 or a+fi—a.fi. But if events are incompossible 





or incapable of happening together, like a clear sky and 
rain, or a new moon and a full moon, then the events are 
not really A or B, but A not-B, or B not- A, or in symbols 
Ab •!• aB. Now if we take /a = probability of A6 and 
V = probability of aB, then we may add simply, and the 
probability of Ab I- aB is /* + v. 

Let the reader carefully observe that if the combi- 
nation AB cannot exist, the probability of Ab is not the 
product of the probabilities of A and b. When certain 
combinations are logically impossible, it is no lonj^er 
allowable to substitute the probability of each term for 
the term, because the multiplication of probabilities pre- 
supposes the independence of the events. A large part of 
Boole's Laws of Thought is devoted to an attempt to 
overcome this difficulty and to produce a Oeneral Method 
in Probabilities by which from certain logical conditions 
and certain given probabilities it would he possible to 
deduce the probability of any other combinations of 
events under those conditions. Boole pursued his task 
with wonderful ingenuity and power, but after spending 
much study on his work, I am compelled to adopt the 
conclusion that his method is fundamentally erroneous. 
As pointed out by Mr. Wilbraham.^ Boole obtained his 
results by an arbitrary assumption, which is only the most 
probable, and not the only possible assumption. Tiie 
answer obtained is therefore not the real probability, 
which is usually indeterminate, but only, as it were, the '■ 
most probable probability. Certain problems solved by. 
Boole are free from logical conditions and therefore may 
admit of valid answers. These, as I have shown,* may be 
solved by the combinations of the Logical Alphabet, but 
the rest of the problems do not admit of a determinate 
answer, at least by Boole's method. 

Comparison of the Huory with Experience. 

The Laws of Probability rest upon the fundamental prin- 
ciples of reasoning, and cannot be really negatived by any 

' Phihtophical Magazine, 4th Scries, vol. vii. p. 465 ; vol. viii 
p. 01. 

Alemoirs of the Manchester Literary and Philosophical Society, 
3rd Senei, vol. iv. p. 347 




fS i^^^Tu""^ ^' '"^''' '•^PPe" tl>«^ a person 
should always throw a coin head up^rmost, and Appear 

Lottti "i f K ^° "^^ ^y «'^'^»'=^ The theory w^d 
not be falsified, because it contemplates the possibiUty of 

might be counter to all that is probable; the whole 
course of events might seem to be'^m complete contra! 
diction to what we should expect, and yet a casual con^ 

illsT Z •1,'^f?'! ""S*"* ^ '^' ^'^ explanation iTk 
ittrih^« i fi "J," '*""*, ^^^"^ coincidences, which we 

Sn.1 /^ ^^"^ °^ "**"'^' """^ <J"« to the accidental 
conjunction of phenomena in the cases to which our 
attention is directed. All that we can learn from finite 
experience is capable, according to the theory of probabili- 

hS ° ^•^^^'"g ««' ""d it is only infinite experience 
that could assure us of any inductive truths 

r,.;^'Ji ?""?, t™«' the probability that any extreme 
runs of luck will occur is so excessively slight that it 
would be absurd seriously to expect their^occSn^nca I 
il^f "P^'f'^i*''/"' "Stance, that any whist player 

S f7i! P'*^"* '" *°y *^» g»°>«« ^here the distri^ 
bution of the cards was exactly the same, by pure accident 

^^'J^P'e V* *^"?8 ^ * 1*'^° ai'-ays losing at 

a game of pure chance, is wholly unknown. Coincidences 

li^l ru w.°°' i"'P'"«iWe. as I have said, but they 
T «•>, ""''^^IJ^ .that the lifetime of any person, or indeed 

mbSv Tr^'^'^'y' ^"^ """^^ive any 'ap^rSble 
probability of their being encountered. Whenever we 
make any extensive series of trials of chance results, as in 
throwing a die or coin, the probability is great th^t the 

7^^ "v '^'*' "''^'y ^'* ^^^ prodictiofs yielded by 
«wZ' I'^'^'' agreement must not be expected, for that, 
as the theory shows, is highly improbSle. Several 
a^.^f fK*"" ^«° ""«le to test, in this way, the accorf- 
W^ffhf"""! f*^ experience. Buflfon caused the first 
l?^J; """*^-^y " ^"'"•S "^^^ ^ho thr^w a coin many 
times in succession, and he obtained 1992 tails to 2048 

iwn ttui P^P'* "' De Moi^n repeated the trial for Us 
own satisfaction, and obtained 2044 tails to 2048 heads. In 
both cases the coincidence with theory is as cltse as could 





Quetelet also tested the theory in a rather more com- 
plete manner, by placing 20 black and 20 white balls in an 
urn and drawing a ball out time after time in an indifferent 
manner, each ball being replaced before a new drawing was 
made. He found, as might be expected, that the greater 
the number of drawings made, the more nearly were the 
white and black balls equal in number. At the ter- 
mination of the experiment he had registered 2066 white 
and 2030 black balls, the ratio being 102.1 

I have made a series of experiments in a third manner, 
which seemed to me even more interesting, and capable 
of more extensive trial Taking a handful of ten coins, 
usually shillings, I threw them up time after time, and 
registered the numbers of heads which appeared each 
time. Now the probability of obtaining 10, 9, 8, 7, &c., 
heads is proportional to the number of combinations of 
10, 9, 8, 7, &c., things out of 10 things. Consequently 
the results ought to approximate "to the numbers in the 
eleventh line of the Arithmetical Triangle. I made 
altogether 2C48 throws, in two sets of 1024 throws each, 
and the numbers obtained are given in the following 
table : — 







xo Heads o Tail 

9 M I .« 

8 >• • *. 
7 M 3 *> 
6 M 4 .. 

5 t» 5 >• 
4 •• " »t 
3 ». 7 » 
» - « .. 

> >* 9 M 
M 10 M 




























+ I 

+ 7* 
+ so 
+ 6 


— XI 

+ 8 






— I 

He whole number of single throws of coins amounted 
to 10 X 2048, or 20480 in all, one half of which or 
10^340 should theoretically give head. The total number 

» Letters on the Theory of FrobabUitits. translated by Downes, 1840. 
PP- 30, 37. 




of heads obtained was actually 10,353, or 5222 in the 
first series, and 5 131 in the second. The coincidence 
with theory is pretty close, but considering the large 
number of throws there is some reason to suspect a 
tendency in favour of heads. 

The special interest of this trial consists in the ex- 
hibition, in a practical form, of the results of Bernoulli's 
theorem, and the law of error or divergence from the 
mean to be afterwards more fully considered. It illus- 
trates the connection between combinations and permu- 
tations, which is exhibited in the Arithmetical Triangle, 
and which underlies many important theorems of scienca' 

Frcbahle Deductive Arguments. 

With the aid of the theory of probabilities, we may 
extend tlie sphere of deductive argument. Hitherto we 
have treated propositions as certain, and on the hypo- 
thesis of certainty have deduced conclusions equally 
^^\^^^' ^^^ *'^ie information on which we reason in 
ordinary life is seldom or never certain, and almost all 
reasoning is reaUy a question of probability. We oufrht 
therefore to be fully aware of the mode and degree^'in 
which deductive reasoning is affected by the theory of 
probability, and many persons may be surprised at the 
results which must be admitted. Some controversial 
writers appear to consider, as De Morgan remarked ^ that 
an inference from several equally probable premises is 
itself as probable as any of them, but the true result is 
very different. If an argument involves many proposi- 
tiODs, and each of them is uncertain, the conclusion will 
be of very little force. 

The validity of a conclusion may be regarded as a com- 
pound event, depending upon the premises happening 
to be true ; thus, to obtain the probability of the conclusion 
we must multiply together the fractions expressing the 
probabilities of the premises. If the probability is k that 
A is B, and also J that B is C, the conclusion that A is C 
on the ground of these premises, is J x ^ or J. Similarly if 
there be any number of premises requisite to the establish- 

' Eneyclopadta Metropolitana, art. ProbabiliUesy i>. 396. 

i u 









ment of a conclusion and their probabilities be p, q, r, &c., 
the probability of the conclusion on the ground of these 

premises isp x q x r x This product kos but a small 

value, unless each of the quantities p, q, &c, be nearly 


But it is particularly to be noticed that the probability 
thus calculated is not the whole probability of the con- 
clusion, but that only which it derives from the premises 
in question. Whately's ^ remarks on this subject might 
mislead the reader into supposing that the calculation is 
coiijpleted by multiplying together the probabilities of the 
premises. But it has been fully explained by De Morgan * 
that we must take into account the antecedent probability 
of the conclusion ; A may be C for other reasons besides 
its being B, and as he remarks, " It is difficult, if not 
impossible, to produce a chain of argument of which the 
rcasoner can rest the result on those arguments only." 
The failure of one argument does not, except under special 
circumstances, disprove the truth of the conclusion it is 
intended to uphold, otherwise there ai-e few truths which 
could survive the ill-considered arguments adduced in their 
favour. As a rope does not necessarily break because one 
or two strands in i fail, so a conclusion may depend upon 
an endless number of considerations besides those imme- 
diately in view. Even ^hen we have no other informa- 
tion we must not consider a statement as devoid of all 
probability. The true expression of complete doubt is a 
ratio of equality between the chances in favour of and 
against it, and this ratio is expressed in the probability J. 

Now if A and C are wholly unknown things, we have 
no reason to believe that A is C rather than A is not C. 
The antecedent probability is then i. If we also have the 
probabilities that A is B, J and that B is C, J we have no 
right to suppose that the probability of A being C is re- 
duced by the argument in its favour. If the conclusion is 
true on its own grounds, the failure of the argument docs 
not affect it ; thus its total probability is its antecedent 
probability, added to the probability that this failing, tlie 
new argument in question establishes it There is a pro- 

» Elements of LogtCj Book III. sections 1 1 and 18. 

• Encyelopadia Mdropoliiana^ art ProhahiliiiUy p. 40a 


bability J that we shall not require the special argument; 
a probability ^ that we shall, and a probability J that the 
argument does in that case establish it. Thus the com- 
plete result is J + ^ X i, or |. In general language, if a 
be the probability founded on a particular ai'gument, and 
c the antecedent probability of the event, the general result 
is I - (i - a) (i - c), or a + c - oc. 

We may put it still more generally in this way : — Let 
a, 6, c, &c. be the probabilities of a conclusion grounded 
on various arguments. It is only when all the arguments 
fail that our conclusion proves finally untrue ; the proba- 
bilities of each failing are respectively, i — a, i — 6, i ~ c, 
&C. ; the probability that they will all fail is (i - a)(i _ h) 
(l - c)... ; therefore the probability that the conclusion 
will not fail is i - (i - a){i - h){i - c)... &c. It follows 
that every argument in favour of a conclusion, however 
flimsy and slight, adds probability to it When it is 
unknown whether an overdue vessel has foundered or not, 
every slight indication of a lost vessel will add some proba- 
bility to the belief of its loss, and the disproof of any 
particular evidence will not disprove the event 

We must apply these principles of evidence with great 
care, and observe that in a great proportion of cases the 
adducing of a weak argument does tend to the disproof 
of its conclusion. The assertion may have in itself great 
inherent improbability as being opposed to other evidence 
or to the supposed law of nature, and every reasoner may 
be assumed to be dealing plainly, and putting forward the 
whole force of evidence which he possesses in its favour. 
If he brings but one argument, and its probability a is 
small, then in the formula i - (i- a)(i - e) both a and c 
are small, and the whole expression has but little value. 
The whole effect of an ai^ument thus turns upon the 
question whether other arguments remain, so that we can 
introduce other factors (1-6), (i -rf), &c., into the above 
expression. In a court of justice, in a publication having 
an express purpose, and in many other cases, it is doubtless 
right to assume that the whole evidence considered to 
have any value as regards the conclusion asserted, is put 

To assign the antecedent probability of any proposition, 
may be a matter of difficulty or impossibility, and one 

P 2 









wiUi which logic and the theory of probability have little 
concern. From the general body of science in our posses- 
sion, we must in each case make the best judgment we 
can. But in the absence of all knowledge the probability 
should be considered = J, for if we make it less than this 
we incline to believe it false rather than true. Thus, before 
we possessed any means of estimating the magnitudes of 
tlie fixed stars, the statement that Sirius was greater than 
the sun had a probability of exactly ^ ; it was as likely that 
it would be greater as that it would be smaller ; and so 
of any other star. This was the assumption which Michell 
made in his admirable speculations.^ It might seem, 
indeed, that as every proposition expresses an agreement, 
and the agreements or resemblances between phenomena 
are infinitely fewer than the differences (p. 44), every pro- 
position should in the absence of other information be 
infinitely improbable. But in our logical system every 
term may be indifferently positive or negative, so that we 
express under the same form as many differences as agree- 
ments. It is impossible therefore that we shoidd have 
any reason to disbelieve rather than to believe a statement 
about things of which we know nothing. We can hardly 
indeed invent a proposition concerning the truth of which 
we are absolutely ignorant, except when we are entirely 
ignorant of the terms used. If I ask the reader to assign 
the odds that a " Platythliptic Coefficient is positive " he 
will hardly see his way to doing so, unless he regard them 
as even. 

The assumption that complete doubt is properly ex- 
pressed by ^ lias been called in question by Bishop Terrot,* 
who proposes instead the indefinite symbol J; and he 
considers that "the d priori probability derived from 
absolute ignorance has no effect upon the force of a 
subsequently admitted probability." But if we grant that 
the probability may have any value between o and i, and 
that every separate value is equally likely, then n and 
I — » are equally likely, and the average is always J. Or 
we may take j) , dp to express the probability that our 

' Philosophical Transactions (1767). Abridg. vol. xil. p. 435. 
^ Tra»$actuyiu 0/ tks Edinburgh Fhilosojpkieal Soeieli^, voL xxi 



estimate concerning any proposition should lie be ween 
p and p + dp. The complete pi-obability of the .proposition 
is then the integral taken between the limits i Jind o, or 
again J. 

Diffkidties of the Theory. 

The theory of probability, though undoubtedly true, 
requires very careful application. Not only is it a branch 
of mathematics in which oversights are frequently com- 
mitted, but it is a matter of great difficulty in many cases, 
to be sure that the formula correctly represents the data 
of the problem. These difficulties often arise from the 
logical complexity of the conditions, which might be, 
perhaps, to some extent cleared up by constantly bearing 
in mind the system of combinations as developed in the 
Indirect Ix>gical Method. In the study of probabilities, 
mathematicians had unconsciously employed logical pro- 
cesses far in advance of those in possession of logicians, 
and the Indirect Method is but the full statement of these 

It is very curious how often the most acute and power- 
ful intellects have gone astray in the calculation of 
probabilities. Seldom was Pascal mistaken, yet he in- 
augurated the science with a mistaken solution.^ Leibnitz 
fell into the extraordinary blunder of thinking that the 
number twelve was as probable a result in the throwing 
of two dice as the number eleven.^ In not a few cases the 
false solution first obtained seems more plausible to the 
present day than the correct one since demonstrated. 
James Bernoulli candidly records two false solutions of a 
problem which he at first thought self-evident ; and he 
adds a warning i against the risk of error, especiidly when 
we attempt to reason on this subject without a rigid 
adherence to methodical rules and symbols. Montmort 
was not free from similar mistakes. D'Alembert con- 
stantly fell into blundei-s, and could not perceive, for 
instance, that the probabilities would be the same when 

* Montucla, FUstoire des MathJmatiqueSf \o\. iii. p. 386. 

■ Leibuitz Opera^ Dutens* Edition, vol. vi part i. p. 217. Tod- 
hunter's History of tlte Theory of Probability, p. 48. To the latter 
work I am indebted for many of the statements in the text. 






coins are thrown successively as when thrown simul- 
taneously. Some men of great reputation, such as 
Ancillon, Moses Mendelssohn, Garve, Auguste Comte,* 
Poinsot, and J. S. MiU,'^ have so far misapprehended the 
theoiy, as to question its value or even to dispute its 
validity. The erroneous statements ahout the theory given 
in the earlier editions of Mill's System of Logic were par- 
tially withdrawn in the later editions. 

Many persons have a fallacious tendency to believe that 
when a chance event has happened several times together 
in an unusual conjunction, it is less likely to happen 
again. D'Alembert seriously held that if head was thrown 
three times running with a coin, tail would more probably 
appear at the next trial.' Bequelin adopted the same 
opinion, and yet there is no reason for it whatever. If 
the event be really casual, what has gone before cannot in 
the slightest degree influence it As a matter of fact, the 
more often a casual event takes place the more likely it is 
to happen again; because there is some slight empirical 
evidence of a tendency. The source of the fallacy is to be 
found entirely in the feelings of surprise with whicli we 
witness an event happening by chance, in a manner which 
seems to proceed from design. 

Misapprehension may also arise from overlooking the 
difference between permutations and combinations. To 
throw ten heads in succession with a coin is no more 
unlikely than to throw any other particular succession 
of heads and tails, but it is much less likely than five 
heads and five tails without regard to their order, be- 
cause there are no less than 252 different particular 
throws which will give this result, when we abstract 
the difference of order. 

Difliculties arise in the application of ,the theory from 
our habitual disregard of slight probabilities. We are 
obliged practically to accept truths as certain which are 
nearly so, because it ceases to be worth while to calculate 
the difference. No punishment could be inflicted if 
absolutely certain evidence of guilt were required, and as 

* Positive Philosophy f translated by Martineau, vol. ii. p. 12a 

■ SvsUm of Loqioy bk. iii. chap. 18, 5th Ed. vol. ii. p. 61. 

■ Montucla, Histaire, vol iii p. 405 ; Todhunter, p. 263. 




Locke remarks, " He that will not stir till he infallibly 
knows the business he goes about will succeed, will 
have but little else to do but to sit still and perish."* 
There is not a moment of our lives when we do not lie 
under a slight danger of death, or some most terrible fate. 
There is not a single action of eating, drinking, sitting 
down, or standing up, which has not proved fatal to some 
person. Sevei-al philosophers have tried to assign the 
Umit of the probabilities which we regard as zero ; Buffon 
named tv.^tttt' because it is the probability, practically 
disregarded, that a man of 56 years of age will die the next 
day. Pascal remarked that a man would be esteemed a 
fool for hesitating to accept death when three dice gave 
sixes twenty times running, if his reward in case of a 
different result was to be a crown ; but as the chance of 
death in question is only i -i- 6^, or unity divided by 
a number of 47 places of figures, we may be said to incur 
greater risks every day for less motives. There is far 
greater risk of death, for instance, in a game of cricket or 
a visit to the rink. 

Nothing is more requisite than to distinguish carefully 
between the truth of a theory and the truthful application 
of the theory to actual circumstances. As a general rule, 
events in nature and art will present a complexity of 
relations exceeding our powers of treatment The intricate 
action of the mind often intervenes and renders complete 
analysis hopeless. If, for instance, the probability that 
a marksman shall hit the target in a single shot be i in 
10, we might seem to have no difficulty in calculating 
the probability of any sucession of hits ; thus the proba- 
bility of three successive hits would be one in a thousand. 
But, in reality, the confidence and experience derived from 
the first successful shot would render a second success 
more probable. The events are not really independent, 
and there would generally be a far greater preponderance 
of runs of apparent luck, than a simple calculation of 
probabilities could account for. In some persons, however, 
a remarkable series of successes will produce a degree of 
excitement rendering continued success almost impossible. 

Attempts to apply the theory of probability to the 


1 Essay concerning Uwman Unier standing y bk. iv. clu 14. § n. 




results of judicial proceedings have proved of little value, 
siraply because the conditions are far too intricate. As 
Laplace said, " Tant de passions, d'int^rets divers et de 
circonstances compliquent les questions relatives d ces 
objets, qu'elles sont presque toujours insolubles." Men 
acting on a jury, or giving evidence before a court, are 
subject to so many complex influences that no mathema- 
tical formulas can be framed to express the real conditions. 
Jurymen or even judges on the bencli cannot be regarded 
as acting independently, with a definite probability in 
favour of each delivering a correct judgment. Each man 
of the jury is more or less influenced by the opinion of the 
others, and there are subtle effects of character and manner 
and strength of mind which defy analysis. Even in 
physical science we can in comparatively few cases apply 
the theory in a definite manner, because the data required 
are too complicated and difficult to obtain. But such failures 
in no way diininish the truth and beauty of the theory 
itself ; in reality there is no branch of science in which our 
symbols can cope with the complexity of Nature. As 
Donkin said, — 

" I do not see on what ground it can be doubted that 
every definite state of belief concerning a proposed hypo- 
thesis, is in itself capable of being represented by a nume- 
rical expression, however difficult or impracticable it may 
be to ascertain its actual value. It would be very difficult 
to estimate in numbera the vis viva of all the particles of 
a human body at any instant ; but no one doubts that it is 
capable of numerical expression." ^ 

The difficulty, in short, is merely relative to our know- 
ledge and skill, and is not absolute or inherent in the 
subject We must distinguish between what is theo- 
retically conceivable and what is practicable with our 
present mental resources. Provided that our aspirations 
are pointed in a right direction, we must not allow them 
to be damped by the consideration that they pass beyond 
what can now be turned to immediate usa In spite of 
its immense difficulties of appliciition, and the aspersions 
which have been mistakenly cast upon it, the theory of 
probabilities, I repeat, is the noblest, as it will in course 

» Philosophical Magazine, 4th Series, vol. i. p. 354 




of time prove, perhaps the most fruitful branch of mathe- 
matical science. It is the very guide of life, and hardly 
can we take a step or make a decision of any kind without 
correctly or incorrectly making an estimation of proba- 
bilities. In the next chapter we proceed to consider how 
the whole cogency of inductive reasoning rests upon pro- 
babilities. The truth or untruth of a natural law, when 
carefully investigated, resolves itself into a high or low 
degree of probability, and this is the case whether or not 
we are capable of producing precise numerical data. 







We have inquired into the nature of perfect induction, 
whereby we pass backwards from certain observed com- 
binations of events, to the logical conditions governing 
such combinations. We have also investigated the grounds 
ot that theory of probability, whicli must be our guide when 
we leave certainty behind, and dilute knowledge with 
Ignorance. There is now before us the difficult task of 
endeavouring to decide how, by the aid of that theory, we 
can ascend from the facts to the laws of nature ; and may 
then with more or less success anticipate the future 
couree of events. All our knowledge of natural objects 
must be ultimately derived from observation, and the 
diflicult question arises— How can we ever know anything 
which we have not directly observed through one of our 
senses, the apertures of the mind ? The utility of reason- 
ing is to assure ourselves that, at a determinate time and 
place or under specified conditions, a certain phenomenon 
wiU be observed. When we can use our senses and per- 
ceive that the phenomenon does occur, reasoning is super- 
fluous If the senses cannot be used, because the event 
18 in the future, or out of reach, how can reasoning take 
their place ? Apparently, at least, we must infer the un- 
known from the known, and the mind must itself create 
an addition to the sum of knowledge. But I hold that it 
is quite impossible to make any real additions to the con- 
tents of our knowledge, except through new impressions 
upon the senses, or upon some aeai, pf feeling. I shall 


attempt to show that inference, whether inductive or 
deductive, is never more than an unfolding of the contents 
of our exx)erience, and that it always proceeds upon the 
assumption that the future and the unperceived will be 
governed by the same conditions as the past and the 
perceived, an assumption which will often prove to be 

In inductive as in deductive reasoning the conclusion 
never passes beyond the premises. Keasoning adds no 
more to the implicit contents of our knowledge, than the 
arrangement of the specimens in a museum adds to the 
number of those specimens. Arrangement adds to our 
knowledge in a certain sense : it allows us to perceive the 
similarities and peculiarities of the specimens, and on the 
assumption that the museum is an adequate representation 
of nature, it enables us to judge of the prevailing forms of 
natural objects. Bacon's first aphorism holds perfectly 
true, that man knows nothing but what he has observed, 
provided that we include his whole sources of experience, 
and the whole implicit contents of his knowledge. In- 
ference but unfolds the hidden meaning of our observations, 
and the theory of probability shows how far we go beyond 
our data in assuming that new specimens will resemble the 
old ones, or that the future may be regarded as proceeding 
unifoiinly with the past. 

Varums Gla,sses of Inductive TtuOls. 

It will be desirable, in the first place, to distinguish 
between the several kinds of truths which we endeavour 
to establish by induction. Although there is a certain 
common and universal element in all our processes of 
reasoning, yet diversity aiises in their application. 
Similarity of condition between the events from which 
we argue, and those to which we argue, must always be 
the ground of inference; but this similarity may have 
regawl either to time or place, or the simple logical 
combination of events, or to any conceivable junction of 
circumstances involving quality, time, and place. Haying 
met with many pieces of substance possessing ductility 
and a bright yellow colour, and having discovered, by 
perfect induction, that they all possess a hifh si>ecific 





gravity, and a freedom from the corrosive action of acids, 
we are led to expect that every piece of substance, possess- 
ing like ductility and a similar yellow colour, will have an 
equally high specific gravity, and a like freedom from 
corrosion by acids. This is a case of the coexistence of 
qualities ; for the character of the specimens examined 
alters not with time nor place. 

In a second class of cases, time will enter as a prin- 
cipal ground of similarity. When we hear a clock 
pendulum beat time after time, at equal intervals, and 
with a uniform sound, we confidently expect that the stroke 
will continue to be repeated uniformly. A comet having 
appeared several times at nearly equal intervals, we infe*r 
that it will probably appear again at the end of another 
like interval. A man who has returned home evening 
after evening for many years, and found his house stand*^ 
mg, may, on like grounds, expect that it Avill be standing 
the next evening, and on many succeeding evenings. Even 
the continuous existence of an object in an unaltered state, 
or the finding again of that which we have hidden, is but 
a matter of inference depending on experience. 

A still larger and more complex class of cases involves 
the relations of space, in addition to those of time and 
quality. Having observed that every triangle drawn upon 
the diameter of a circle, with ite apex upon the circum- 
ference, apparently contains a right angle, we may 
ascertain that all triangles in similar circumstances will 
contain right angles. This is a case of pure space reason- 
ing, apart from circumstances of time or quality, and it 
seems to be governed by different principles of reasoning. 
I shall endeavour to show, however, that geometrical 
reasoning differs but in degree from that which applies 
to other natural relations. 

The Relation of Cause and Effect. 

In a very large part of the sci^tific investigations 
which must be considered, we deal with events which 
follow from previous events, or with existences which 
succeed existences. Science, indeed, might arise even were 
material nature a fixed and changeless whole. Endow 
mind with the power to travel about, and compare part 


with part, and it could certainly draw inferences concern- 
ing the similarity of forms, the coexistence of qualities, 
or^the preponderance of a particular kind of matter in 
a changeless world. A solid universe, in at least approxi- 
mate equilibrium, is not inconceivcvble, and then the rela- 
tion of cause and effect would evidently be no more than 
the relation of before and after. As nature exists, how- 
ever, it is a progressive existence, ever moving and 
changing as time, the great independent variable, pro- 
ceeds. Hence it arises that we must continually compare 
what is happening now with what happened a moment 
befoi-e, and a moment before that moment, and so on, 
until we reach indefinite periods of past time. A comet 
is seen moving in the sky, or its constituent particles 
illumine the heavens with their tails of fire. We cannot 
explain the present movements of such a body without 
supposing its prior existence, with a definite amount 
of energy and a definite direction of motion ; nor can we 
validly suppose that our task is concluded when we find 
that it came wandering to our solar system through the 
unmeasured vastness of surrounding space. Every event 
must have a cause, and that cause again a cause, until 
wo are lost in the obscurity of the past, and are driven to 
the belief in one First Cause, by whom the course of 
nature was determined. 

Fallacious Use of the Term Cause. 

Tlie words Cause and Causation have given rise to infinite 
trouble and obscurity, and have in no slight degree retarded 
the progress of science. From the time of Aristotle, the 
work of philosophy has been described as the discovery of 
the causes of things, and Francis Bacon adopted the notion 
when he said " vere scire esse per causas scire." Even now 
it is not uncommonly supposed that the knowledge of 
causes is something different from other knowledge, and 
consists, as it were, in getting possession of the keys of 
nature. A singje word may thus act as a spell, and throw 
the clearest intellect into confusion, as I have often thought 
that Locke was thrown into confusion when endeavouring 
to find a meaning for the word power} In Mill's System of 

^ Et$ay caticemtng Human Undentanding, bk. iL chap, xxi 







Logic the term catise seems to have re-asserted its old 
noxious power. Not only does Mill treat the Laws of 
Causation as almost coextensive with science, but he so 
uses the expression as to imply that when once we pass 
wiUun the circle of causation we deal with certainties. 

The philosophical danger which attaches to the use of 
this word may be thus described. A cause is defined as 
the necessary or invariable antecedent of an event so 
that when the cause exists the effect wiU also exist or 
soon follow. If then we know the cause of an event we 
know what will certainly happen ; and as it is implied 
that science, by a proper experimental method, may attain 
to a knowledge of causes, it follows that experience may 
give us a certain knowledge of future events. But nothing 
IS more unquestionable than that finite experience can 
never give us certain knowledge of the future, so that 
either a cause is not an invariable antecedent, or else we 
can never gain certain knowledge of causes. The first 
horn of this dilemma is hardly to be accepted. Doubtless 
there IS m nature some invariably acting mechanism, such 
that from certain fixed conditions an invariable result 
always emerges. But we, with our finite minds and 
short expenence, can never penetrate the mystery of 
those existences which embody the Will of the Creator 
and evolve it throughout time. We are in the position 
ot spectators who witness the productions of a compli- 
cated machine, but are not aUowed to examine its inti- 
mate structure. We learn what does happen and what 
does appear, but if we ask for the reason, the answer 
would involve an infinite depth of mystery. The simplest 
bit of matter, or the most trivial incident, such as the 
stroke of two billiard balls, offers infinitely more to learn 
than ever the human intellect can fathom. The word 
cause covers just as much untold meaning as any of the 
words siibstancey matter, tkougJU, existence. 

Confusion of Two Questions, 

The subject is much complicated, too, by the confusion 
of two distinct questions. An event having happened, we 
may asK—"* 


(i) Is there any cause for the event ? 

(2) Of what kind is that cause ? 
No one would assert that the mind possesses any 
faculty capable of inferring, prior to experience, that the 
occurrence of a sudden noise with flame and smoke indi- 
cates the combustion of a black powder, formed by the 
mixture of black, white, and yellow powders. The greatest 
upholder of d priori doctrines will allow that the parti- 
cular aspect, shape, size, colour, texture, and other 
qualities of a cause must be gathered through the senses. 

The question whether there is any cause at all for an 
event, is of a totally different kind. If an explosion could 
happen without any prior existing conditions, it must be 
a new creation — a distinct addition to the universe. It 
may be plausibly held that we can imagine neither the 
creation nor annihilation of anything. As regards matter, 
this has long been held true ; as regards force, it is now 
almost universally assumed as an axiom that energy can 
neither come into nor go out of existence without distinct 
acts of Creative Will. That there exists any instinctive 
belief to this effect, indeed, seems doubtful. We find 
Lucretius, a philosopher of the utmost intellectual power 
and cultivation, gravely assuming that his raining atoms 
could turn aside from their straight paths in a self-deter- 
mining manner, and by this spontaneous origination of 
energy determine the form of the universe.^ Sir George 
Airy, too, seriously discussed the mathematical conditions 
under which a perpetual motion, that is, a perpetual 
source of self-created energy, might exist.* The larger 
part of the philosophic world has long held that in mental 
acts there is free will — in short, self-causation. It is in 
vain to attempt to reconcile this doctrine with that of an 
intuitive belief in causation, as Sir W. Hamilton candidly 

It is obvious, moreover, that to assert the existence 
of a cause for every event cannot do more than remove 
into the indefinite past the inconceivable fact and mystery 
of creation At any given moment matter and energy 

^ De i?«rt,m NcUuraj bk. ii. IL 216-293. 

* Cambr%d{fe Fhiloiophieal Transaetumt (1830), yol 


lu. p|». 




I < 


were equal to wliat they are at present, or they were 
not ; if equal, we may make the same inquiry concerning 
any other moment, however long prior, and we are thus 
obliged to accept one horn of the dilemma — existence 
from infinity, or creation at some moment. This is but 
one of the many cases in which we are compelled to believe 
in one or other of two alternatives, both inconceivable. 
My present purpose, however, is to point out that we must 
not confuse this supremely diflBcult question with that 
into which inductive science inquires on the foundation of 
facts. By induction we gain no certain knowledge ; but 
by observation, and the inverse use of deductive reasoning, 
we estimate the probability that an event which has 
occurred was preceded by conditions of specified character, 
or that such conditions will be followed by the event. 

Definition of the Term Cause, 

Clear definitions of the word cause have been given by 
several philosophers. Hobbes has said, " A cause is the 
sum or aggregate of all such accidents, both in the agents 
and the patients, as concur in the producing of the effect 
propounded ; all which existing together, it cannot be 
understood but that the effect existeth with them; or 
that it can possibly exist, if any of them be absent." 
Brown, in his Essay on Causation, gave a nearly corre- 
sponding statement. "A cause," he says,* "may be 
defined to be the object or event which immediately 
precedes any change, and which existing again in similar 
circumstances will be always immediately followed by a 
similar change." Of the kindred word power, he like- 
wise says : * *' Power is nothing more than that invariable- 
ness of antecedence which is implied in the belief of 

These definitions may be accepted with the qualifica- 
tion that our knowledge of causes in such a sense can be 
probable only. The work of science consists in ascertaining 
the combinations in which phenomena present themselves. 

* Observaiumi on the Nature and Tendency of the Doctrine of 
Hr. Hume, concerning the Relation of Cause and Effect, Second ed. 
^ 44. * Ibid. p. 97. 


Concerning every event we shall have to determine its 
probable conditions, or the group of antecedents from which 
it probably follows. An antecedent is anything which 
exists prior to an event; a consequent is anything which 
exists subsequently to an antecedent. It will not usually 
happen that there is any probable connection between an 
antecedent and consequent. Thus nitrogen is an antece- 
dent to the lighting of a common fire ; but it is so far from 
being a cause of the lighting, that it renders the combustion 
less active. Daylight is an antecedent to all fires lighted 
during the day, but it probably has no appreciable effect 
upon their burning. But in the case of any given event it 
is usually possible to discover a certain number of ante- 
cedents which seem to be always present, and with more 
or less probability we conclude that when they exist the 
event will follow. 

Let it be observed that the utmost latitude is at present 
enjoyed in the use of the term cause. Not only may a 
cause be an existent thing endowed with powers, as 
oxygen is the cause of combustion, gunpowder the cause 
of explosion, but the very absence or removal of a thing 
may also be a cause. It is quite correct to speak of the 
.dryness of the Egyptian atmosphere, or the absence of 
moistui'e, as being the cause of tlie preservation of 
mummies, and other remains of antiquity. The cause of 
a mountain elevation, Ingleborough for instance, is the 
excavation of the surrounding valleys by denudation. It 
is not so usual to speak of the existence of a thing at one 
moment as the cause of its existence at the next, but to 
me it seems the commonest case of causation which can 
occur. The cause of motion of a billiard ball may be the 
stroke of another ball ; and recent philosophy leads us to 
look upon all motions and changes, a.s but so many mani- 
festations of prior existing energy. In all probability 
there is no creation of energy and no destruction, so that 
as regards both mechanical and molecular changes, the 
cause is really the manifestation of existing energy. In 
the same way I see not why the prior existence of matter 
is not also a cause as regards its subsequent existence. All 
science tends to show us that the existence of the universe 
in a particular state at one moment, is the condition of its 
existence at the next moment, in an apparently different 

f| fl 




state. When we analyse the meaning which we can 
attribute to the word cause^ it amounts to the existence of 
suitable portions of matter endowed with suitable quan-. 
tities of energy. If we may accept Home Tooke's asser- 
tion, cause has etymologically the meaning of thing hefore. 
Though, indeed, the origin of the word is very obscure, its 
derivatives, the Italian cosa, and the French chose^ mean 
simply thing. In the German equivalent ursache, we have 
plainly the original meaning of thing before, the sache 
denoting "interesting or important object," the English 
sake, and tir being the equivalent of the English «rf, 
he/ore. We abandon, then, both etymology and philo- 
sophy, when we attribute to the laws of causation any 
meaning beyond that of the conditions under which an 
event may be expected to happen, according to our 
observation of the previous course of nature. 

I have no objection to use the words cause and 
causation, provided they are never allowed to lead us to 
imagine that our knowledge of nature can attain to cer- 
tainty. I repeat that if a cause is an invariable and 
necessary condition of an event, we can never know 
certainly whether the cause exists or not. To us, then, a 
cause is not to be distinguished from the group of positive 
or negative conditions which, with more or less probability, 
precede an event. In this sense, there is no particular 
difference between knowledge of causes and our general 
knowledge of the succession of combinations, in which the 
phenomena of nature are presented to us, or found to 
occur in experimental inquiry. 

Distinction of Inductive and Deductive Results. 

We must carefully avoid confusing together inductive 
investigations which terminate in the establishment of 
general laws, and those which seem to lead directly to 
the knowledge of future particular events. That process 
only can be called induction which gives general laws, 
and it is by the subsequent employment of deduction that 
we anticipate particular events. If the observation of a 
number of cases shows that alloys of metals fuse at lower 
tempemtures than their constituent metals, I may with 
more or less probability draw a general inference to that 


effect, and may thence deductively ascertain the proba- 
bility that the next alloy examined will fuse at a lower 
temperature than its constituents. It has been asserted, 
indeed, by Mill,^ and partially admitted by Mr. Fowler,^ 
that we can argue directly from case to case, so that what 
is true of some alloys will be true of the next. Professor 
Bain has adopted the same view of reasoning. He thinks 
that Mill has extricated us from the dead lock of the 
syllogism and effected a total revolution in logic. He 
holds that reasoning from particulars to particulars is not 
only the usual, the most obvious and the most ready 
method, but that it is the type of reasoning which best 
discloses tJie real process.' Doubtless, this is the usual 
result of our reasoning, regard being had to degrees o! 
probability ; but these logicians fail entirely to give any 
explanation of the process by which we get from case 
to case. 

It may be allowed that the knowledge of future par- 
ticular evente is the main purpose of our investigations, 
and if there were any process of thought by which we 
could pass directly from event to event without ascending 
into general truths, this method would be sufficient, and 
certainly the briefest. It is true, also, that the laws of 
mental association lead the mind always to expect the like 
again in apparently like circumstances, and even animals 
of very low intelligence must have some trace of such 
powers of association, serving to guide them more or less 
correctly, in the absence of true reasoning faculties. But 
it is the purpose of logic, according to Mill, to ascertain 
whether inferences have been correctly drawn, rather than 
to discover them.* Even if we can, then, by habit, 
association, or any rude process of inference, infer the 
future directly from the past, it is the work of logic to 
analyse the conditions on which the correctness of this 
inference depends. Even Mill would admit that sucl 
analysis involves the consideration of general truths,* am' 

• System of LogiCf bk. II. chap. iii. 

• Inductive Logic, pp. 13, 14. 

• Bain, Deductive Logic, pp. 208, 209. 

• System of Logic. Introduction, § 4. Fifth ed. pp. 8, 9. 

• Ibid. bk. II. chap. iii. } 5, pp. 225, &c 

Q 2 






in this, as iu several other impoilaut points, we might 
controvert Mill's own views by his own statements. It 
seems to me undesirable in a systematic work like this to 
enter into controversy at any length, or to attempt to refute 
the views of other logicians. But I shall feel bound to 
state, in a separate publication, my very deliberate opinion 
that many of Mill's innovations in logical science, and 
especially his doctrine of reasoning from particulars to 
particulars, are entirely groundless and false. 

The Grounds of Iriductive Inference. 

I hold that in all cases of inductive inference we must 
invent hypotheses, until we fall upon some hypothesis 
which yields deductive results in accordance with experi- 
ence. Such accordance renders the chosen hypothesis 
more or less probable, and we may then deduce, with some 
degree of likelihood, the nature of our future experience, 
on the assumption that no arbitrary change takes place in 
the conditions of nature. We can only argue from the 
past to the future, on the general principle set forth in this 
work, that what is true of a thing will be true of the like. 
So far then as one object or event differs from another, all 
inference is impossible, particulars as particulars can no 
moi-e make an infei*ence than grains of sand can make a 
rope. We must always rise to something which is general 
or same in the cases, and assuming that sameness to be 
extended to new cases we learn their nature. Hearing a 
clock tick five thousand times without exception or varia- 
tion, we adopt the very probable hypothesis that there is 
some invariably acting machine which produces those uni- 
form sounds, and which will, in the absience of change, go 
on producing them. Meeting twenty times with a bright 
yellow ductile substance, and finding it always to be \%vy 
heavy and incorrodible, I infer that there was some natural 
condition which tended in the creation of things to asso- 
ciate these properties together, and I expect to find them 
associated in the next instance. But there always is the 
possibility that some unknown change may take place 
between past and future cases. The clock may run down, 
or be subject to 'jl hundred accidents altering its condition. 
TLer^ is no reason in the nature of things, so far as known 


to us, why yellow colour, ductility, high specific gravity, 
and incorrodibility, should always be associated together, 
and in other cases, if not in this, men's expectations 
have been deceived. Our inferences, therefore, always 
retain more or less of a hypothetical character, and are so 
far open to doubt. Only in proportion as our induction 
approximates to the character of j)erfect induction, does 
it approximate to certainty. The amount of uncertainty 
corresponds to the probability that other objects than 
those examined may exist and falsify our inferences ; the 
amount of probability corresponds to the amount of infor- 
mation yielded by our examination ; and the theory of 
probability will be needed to prevent us from over-esti- 
mating or under- estimating the knowledge we possess. 

Illustrations of the Inductive Process, 

To illustrate the passage from the known to the ap- 
parently unknown, let us suppose that the phenomena 
under investigation consist of numbers, and that the 
following six numbers being exhibited to us, we are 
required to infer the character of the next in the 
series : — 

5» i5» 35, 45, 6s, 95. 
The question first of all arises. How may we describe this 
series of numbers ? What is uniformly true of them ? 
The reader cannot fail to perceive at the first glance that 
they all end in five, and the problem is, from the pi-opcr- 
ties of these six numbers, to infer the properties of the 
next number ending in five. If we test their properties 
by the process of perfect induction, we soon perceive that 
they have another common property, namely that of being 
divisible hyflve without remainder. May we then assert that 
the next number ending in five is also divisible by five, 
and, if so, upon what grounds ? Or extending th(^ question, 
Is every number ending in five divisible by five ? Does it 
follow that because six numbers obey a supposed law, 
therefore 376,685,975 or any other number, however large, 
obeys the law ? I answer certainly not. The law in ques- 
tion is undoubtedly true ; but its truth is not proved by 
any finite number of examples. All that these six numbers 
can do is to suggest to my mind the possible existence of 





such a law ; and I then ascertain its truth, by proving 
deductively from the rules of decimal numeration, that any 
number ending in five must be made up of multiples of 
five, and must therefore be itself a multiple. 

To make this more plain, let the reader now examine 
the numbers — 

7> 17* 37, 47, 67, 97. 

They all end in 7 instead of 5, and though not at equal 
intervals, the intervals are the same as in the previous 
case. After consideration, the reader will perceive that 
these numbers all agree in being prime numbers, or mul- 
tiples of unity only. May we then infer that the next, or 
any other number ending in 7, is a prime number? 
Clearly not, for on trial we tind that 27, 57, 117 are not 
primes. Six instances, then, treated empirically, lead us 
to a true and universal law in one case, and mislead us in 
another case. We ought, in fact, to have no confidence in 
any law until we have treated it deductively, and have 
shown that from the conditions supposed the results ex- 
pected must ensue. No one can show from the principles 
of number, that numbers ending in 7 should be primes. 

From the history of the theory of numbers some good 
examples of false induction can be adduced. Taking the 
following series of prime numbers, 

4i,43>47, 53,61,71, 83.97, "3, 131, 151, &c., 
it will be found that they all agree in being values of 
the general expression x* + a? + 41, putting for a; in succes- 
sion the values, o, i, 2, 3, 4, &c. We seem always to 
obtain a prime number, and the induction is apparently 
strong, to the effect that this expression always will 
give piimes. Yet a few more trials disprove this false con- 
clusion. Put X = 40, and we obtain 40 x 40 + 40 + 41, 
or 41 X 41. Such a failure could never have happened, 
had we shown any deductive reason why a;^ + a; + 41 
should give primes. 

There can be no doubt that what here happens with 
forty instances, might happen with forty thousand or 
forty million instances. An apparent law never once 
failing up to a certain point may then suddenly break 
down, so that inductive reasoning, as it has been described 
Dy some writers, can give no sure knowledge of what is to 
come. Babbage pointed out in his Ninth Bridge water 

Treatise, that a machine could be constructed to give a 
perfectly regular series of numbers through a vast series 
of steps, and yet to break the law of progression suddenly 
at any required point. No number of particular cases as 
particulars enables us to pass by inference to any new case. 
It is hardly needful to inquire here what can be inferred 
from an infinite series of facts, because they are never 
practically within our power ; but we may unhesitatingly 
accept the conclusion, that no finite number of instances 
can ever prove a general law, or can give us certain know- 
ledge of even one other instance. 

General mathematical theorems have indeed been dis- 
covered by the observation of particular cases, and may 
again be so discovered. We have Newton's own state- 
ment, to the effect that he was thus led to the all-impor- 
tant Binomial Theorem, the basis of the whole structure 
of mathematical analysis. Speaking of a certain series of 
terms, expressing the area of a circle or hyperbola, he says : 
" I reflected that the denominators were in arithmetical 
progression; so that only the numerical co-efficients of 
the numerators remained to be investigated. But these, 
in the alternate areas, were the figures of the powers of 
the number eleven, namely 11°, 11 \ ii*, ns^ ii*j that is, 
in the first l ; in the second I, i ; in the third i, 2, i ; in 
the fourth i, 3, 3, I ; in the fifth i, 4, 6, 4, i.^ I inquired, 
therefore, in what manner all the remaining figures could 
be found from the fii-st two ; and I found that if the first 
figure be called m, all the rest could be found by the 
continual multiplication of the terms of the formula 

«— o 


^ m-_2 ^ m- 

X &a"« 

3 ' 4 

It is pretty evident, from this most interesting statement, 
that Newton, having simply observed the succession of the 
numbers, tried various formulae until he found one which 
agreed with them alL He was so little satisfied with this 
process, however, that he verified particular results of his 
new theorem by comparison with the results of common 

* These are the figurate numbers considered in pages 183, 187, &c 
' Commercium J^istolicum. Einttola ad Oldenburgum, Oct. 24, 
1676. Horsley's Works of Newton, vol. iv. p. 541. See De Morgan 
in Penny Cvclovcedia art ** Binofnial Theorem," p. 412. 




1 < 

multiplication, and the rule for the extraction of the 
square root. Newton, in fact, gave no demonstration 
of his theorem ; and the greatest mathematicians of the 
last century, James Bernoulli, Maclaurin, Landen, Euler, 
Lagrange, &c., occupied themselves with discovering a con- 
clusive method of deductive proof. 

There can be no doubt that in geometiy also discoveries 
have been suggested by direct observation. Many of the 
now trivial propositions of Euclid's Elements were pro- 
bably thus discovered, by the ancient Greek geometers ; 
and we have pretty clear evidence of this in the Commen- 
taries of Proclus.^ Galileo was the first to examine the 
remarkable properties of the cycloid, the curve described by 
a point in the circumference of a wheel rolling on a plane. 
By direct observation he ascertained that the area of the 
curve is apparently three times that of the generating circle 
or wheel, but he was unable to prove this exactly, or to 
verify it by strict geometrical reasoning. Sir George Airy 
has recorded a curious case, in which he fell accidentally by 
trial on a new geometrical property of the sphere.* But 
discovery in such cases means nothing more than sugges- 
tion, and it is always by pure deduction that the general 
law is really established. As Proclus puts it, "ve must 
pass from sense to consideration. 

Given, for instance, the series of figures in the accom- 
panying diagram, measurement will show that the curved 
lines approxim te to semicircles, and the rectilinear figures 
to right-angled triangles. These figures may seem to 
suggest to the mind the general law that angles inscribed 

* Bk. ii. chap. iv. 

* PhiloMphiccU Transactions (i866), vol. 146, p. 534. 



in semicircles are right angles ; but no number of instances, 
and no possible accuracy of measurement would really 
establish the truth of that general law. Availing ourselves 
of the suggestion furnished by the figures, we can only 
investigate deductively the consequences which flow from 
the definition of a circle, until we discover among them the 
property of containing right angles. Persons have thought 
that they had discovered a method of trisecting angles by 
plane geometrical construction, because a certain complex 
arrangement of lines and circles had appeared to trisect an 
angle in every case tried by them, and they inferred, by a 
supposed act of induction, that it would succeed in all 
other cases. De Morgan has recorded a proposed mode of 
trisecting the angle which could not be discriminated by 
the senses from a true general solution, except when it was 
applied to very obtuse angles.^ In all such cases, it has 
always turned out either that the angle was not trisected 
at all, or that only certain particular angles could be thus 
trisected. The trisectors were misled by some apparent or 
special coincidence, and only deductive proof could es- 
tablish the truth and generality of the result. In this par- 
ticular case, deductive proof shows that the problem 
attempted is impossible, and that angles generally cannot 
be trisected by common geometrical methods. 

Geometrical Reasoning. 

This view of the matter is strongly supported by the 
further consideration of geometrical reasoning. No skill 
and care could ever enable us to verify absolutely any one 
geometrical proposition. Kousseau, in his Umile, tells us 
that we should teach a child geometry by causing him to 
measure and compare figures by superposition. While a 
child was yet incapable of general reasoning, this would 
doubtless be an instructive exercise ; but it never could 
teach geometry, nor prove the truth of any one proposition. 
All our figures are rude approximations, and they may 
happen to seem unequal when they should be equal, 
and equal when they should be unequal Moreover 
figures may from chance be equal in case after case, and 

' Budget of Paradoxes^ p. 257. 





yet there may be no general reason why they should be 
so. The results of deductive geometrical reasoning are 
absolutely certain, and are either exactly true or capable 
of bemg carried to any required degree of approximation 
In a perfect tnangle, the angles must be equal to one half- 
revolution precisely; even an infinitesimal divergence 
would be impossible; and I believe with equal confidence 
that however many are the angles of a figure, provided 
there are no re-entrant angles, the sum of the angles will 
be precisely and absolutely equal to twice as many right- 
angles as the figure has sides, less by four right-aufrles. 
In such cases, the deductive proof is absolute and com- 
plete ; empirical verification can at the most guard against 
accidental oversights. 

There is a second class of geometrical truths which can 
only be proved by approximation ; but, as the mind sees 
no reason why that approximation should not always go 
on, we arrive at complete conviction. We thus learn that 
the surface of a sphere is equal exactly to two-thirds of 
the whole surface of the circumscribing cylinder, or to four 
times the area of the generating circle. The area of a 
parabola is exactly two- thirds of that of the circumscribing 
parallelogram. The area of the cycloid is exactly three 
times that of the generating circle. These are truths that 
we could never ascertain, nor even verify by observation 
for any finite amount of difference, less than what the 
senses can discern, would falsify them. 

There are geometrical relations again which we cannot 
assign exactly, but can cany to any desirable degree of ap- 
proximation. The ratio of the circumference to the dia- 
meter of a circle is that of 314159265358979323846 

to I, and the approximation may be carried to any ex- 
tent by the expenditure of sufficient labour. Mr. W 
Shanks has given the value of this natural constant, known 
as TT, to the extent of 707 places of decimals.^ Some years 
since, I amused myself by trying how near I could get to 
this ratio, by the careful use of compasses, and I did not 
come nearer than i part in 540. We might imagine mea- 
surements so accurately executed as to give us eight or 
ten places correctly. But the power of the hands &mi 

» Froeeedings 0/ the Royal Society (1872-3), yoI. xxi. p. 319. 




senses must soon stop, whereas the mental powers of de- 
ductive reasoning can proceed to an unlimited degree of ap- 
proximation. Geometrical truths, then, are incapable of 
verification ; and, if so, they cannot even be learnt by 
observation. How can I have learnt by observation a pro- 
position of which I cannot even prove the truth by obser- 
vation, when I am in possession of it ? All that observa- 
tion or empiiical trial can do is to suggest propositions, of 
which the truth may afterwards be proved deductively. 

If Viviani's story is to be believed, Galileo endeavoured 
to satisfy himself about the area of the cycloid by cutting 
out several large cycloids in pasteboard, and then compar- 
ing the areas of the curve and the generating circle by 
weighing them. In every trial the curve seemed to be 
rather less than three times the circle, so that Galileo, we 
are told, began to suspect that the ratio was not precisely 
3 to I. It is quite clear, however, that no process of 
weighing or measuring could ever prove truths like these, 
and it remained for Torricelli to show what his master 
Galileo had only guessed at.^ 

Much has been said about the peculiar certainty of 
mathematical reasoning, but it is only certainty of deduc- 
tive reasoning, and equal certainty attaches to all correct 
logical deduction. If a triangle be right-angled, the 
squai-e on the hypothenuse will undoubtedly equal the 
sum of the two squares on the other sides ; but I can 
never be sure that a triangle is right-angled : so I can be 
certain that nitric acid will not dissolve gold, provided I 
know that the substances employed really correspond to 
those on which I tried the experiment previously. Here 
is like certainty of inference, and like doubt as to the 

Discriminatian of Certainty and Probability, 

We can never recur too often to the truth that our 
knowledge of the laws and future events of the external 
world is only probable. The mind itself is quite capable 
of possessing certain knowledge, and it is well to discri- 
minate carefully between what we can and cannot know 

* Life of (JaliUOf Society for the Diffusion of Useful Knowledge, 
p. 102. 





I. '. 


\vith certainty. In the first place, wliatcver feeling is 
actually present to the mind is certainly known to that 
mind. If I see blue sky, I may be quite sure that I 
do experience the sensation of bhieness. Whatever I do 
feel, I do feel beyond all doubt. We are indeed very 
likely to confuse what we really feel with what we are 
inclined to associate with it, or infer inductively from 
it; but the whole of our consciousness, as far as it is 
the result of pure intuition and free from inference, is 
certain knowledge beyond all doubt. 

In the second place, we may have certainty of inference ; 
the fundamental laws of thought, and the rule of substitution 
(p. 9), are cei-tainly true ; and if my senses could inform me 
that A was indistinguishable in colour from B, and B from 
C, then I should be equally cerfain that A was indistinguish- 
able from C. In short, whatever tnith there is in the 
premises, I can certainly embody in their correct logical 
result. But the certainty generally assumes a hypothetical 
character. I never can be quite sure that two colours 
are exactly alike, that two magnitudes are exactly equal, 
or that two bodies whatsoever are identical even in their 
apparent qualities. Almost all our judgments involve 
quantitative relations, and, as will be shown in succeeding 
chaptei-s, we can never attain exactness and certainty 
where continuous quantity enters. Judgments concerning 
discontinuous quantity or numbers, however, allow of cer- 
tainty ; I may establish beyond doubt, for instance, that 
the difference of the squares of ly and 13 is the product 
of 17 + 13 and 17—13, and is therefore 30 x 4, or 120. 

Inferences which we draw concerning natural objects 
are never certain except in a hypothetical point of 
view. It might seem to be certain that iron is magnetic, 
or that gold is incapable of solution in nitric acid ; but, 
if we carefully investigate the meanings of these state- 
ments, they will be found to involve no certainty but 
that of consciousness and that of hypothetical inference. 
For what do I mean by iron or gold? If I choose a 
remarkable piece of yellow substance, call it gold, and 
then immerse it in a liquid which I call nitric acid, and 
find that there is no change called solution, then conscious- 
ness has certainly informed me that, with my meaning of 
the terms, '* Grold is insoluble in nitric acid." I may further 



be certain of something else ; for if this gold and nitric 
acid remain what they were, 1 may be sure there will be 
no solution on again trying the experiment. If I take other 
portions of gold and nitric acid, and am sure that they really 
are identic«d in properties with the former portions, I can 
be certain that there will be no solution. But at this point 
my knowledge becomes purely hypothetical ; for how can I 
be sure without trial that the gold and acid are really 
identical in nature with what I formerly called gold and 
nitric acid. How do I know gold when I see it ? If 1 
judge by the apparent qualities — coloui-, ductility, specific 
gravity, &c., I may be misled, because there may always 
exist a substance which to the colour, ductility, specific 
gravity, and other specified qualities, joins others which we 
do not expect Similarly, if iron is magnetic, as shown by 
an experiment with objects answering to those names, then 
all iron is magnetic, meaning all pieces of matter identical 
with my assumed piece. But in trying to identify iron, I 
am always open to mistake. Nor is this liability to mis- 
take a matter of speculation only.^ 

The history of chemistry shows that the most confident 
inferences may have been falsified by the confusion of one 
substance with another. Thus strontia was never discri- 
minated from baryta until Klapruth and Hauy detected 
differences between some of their properties. Accordingly 
chemists must often have inferred concerning strontia 
what was only true of baryta, and vice versd. There is 
now no doubt that the recently discovered substances, 
caesium and rubidium, were long mistaken for potassium.^ 
Other elements have often been confused together — for 
instance, tantalum and niobium ; sulphur and selenium ; 
cerium, lanthanum, and didymium ; yttrium and erbium. 

Even the best known laws of physical science do 
not exclude false inference. No law of nature has been 
better established than that of universal gravitation, and 
we believe with the utmost confidence that any body 
capable of affecting the senses will attract other bodies, 
and fall to the earth if not prevented. Euler remarks 


1 Professor Bowen has excellently stated this view. 
Logic Cambridge, U.S.A., 1866, p. 354. 
^ BoBcoe's Spectrum Anuiysis, ist edit., p. 98. 

Treatise on 






tliat, although he had never made trial of the stones 
which compose the church of Magdeburg, yet he had 
not the least doubt that all of them were heavy, and 
would fall if unsupported. But he adds, that it would 
be extremely difficult to give any satisfactory explanation 
of this confident belief.^ The fact is, that the belief ought 
not to amount to certainty until the experiment has been 
tried, and in the meantime a slight amount of uncer- 
tainty enters, because we cannot be sure that the stones of 
the Magdeburg Church resemble other stones in all their 

In like manner, not one of the inductive truths which 
men have established, or think they have established, is 
really safe from exception or reversal. Lavoisier, when 
laying the foundations of chemistry, met with so many 
instances tending to show the existence of oxygen in 
all acids, that he adopted a general conclusion to that 
effect, and devised the name oxygen accordingly. He 
entertained no appreciable doubt that the acid existing 
in sea salt also contained oxygen;* yet subsequent ex- 
perience falsified his expectations. This instance refers 
to a science in its infancy, speaking relatively to the 
possible achievements of men. But all sciences are and 
ever will remain in their infancy, relatively to the extent 
and complexity of the universe which they undertake to 
investigate. Euler expresses no more than the truth when 
he says that it would be impossible to fix on any one thing 
really existing, of which we could have so perfect a know- 
ledge as to put us beyond the reach of mistake." We may 
be quite certain that a comet will go on moving in a 
similar path if all circumstances remain the same as 
before ; but if we leave out this extensive qualification, 
our predictions will always be subject to the chance of 
falsification by some unexpected event, such as the division 
of Biela's comet or the interference of an unknown gravi- 
tating body. 


Inductive inference might attain to certainty if our 
knowledge of the agents existing throughout the universe 
were complete, and if we were at the same time certain 
tliat the same Power which created the universe would 
allow it to proceed without arbitrary change. There is 
always a possibility of causes being in existence without 
our knowledge, and these may at any moment produce 
an unexpected effect. Even when by the theory of pro- 
babilities we succeed in forming some notion of the com- 
parative confidence with which we should receive in- 
ductive results, it yet appears to me that we must make 
an assumption. Events come out like balls from the vast 
ballot-box of nature, and close observation wiU enable us 
to form some notion, as we shall see in the next chapter, 
of the contents of that ballot-box. But we must still 
assume that, between the time of an observation and that 
to which our infei-ences relate, no change in the ballot-box 
has been made. 

* Euler's Letten to a Oerman Priw«M, translated by Hunter. 
2nd ed., vol. ii. pp. 17, 18. 

2 Lavoisier's Chemistry ^ translated by Kerr. 3id ed., pp. X14, 
lai, 123. 

* Euler's Letters, vol. il p. 21. 

I ) 




We have hitherto considered the theory of probability 
only in its simple deductive employment, in which it 
enables us to determine from given conditions the probable 
character of events happening under those conditions. 
But as deductive reasoning when inversely applied con- 
stitutes the process of induction, so the calculation of 
probabilities may be inversely applied ; from the known 
character of certain events we may argue backwards to 
the probability of a certain law or condition governing 
those events. Having satisfactorily accomplished this 
work, we may indeed calculate forwards to the probable 
character of future events happening under the same con- 
ditions ; but this part of the process is a direct use of 
deductive reasoning (p. 226). 

Now it is highly instructive to find that whether the 
theoiy of probability be deductively or inductively ap- 
plied, the calculation is always performed according to 
the principles and rules of deduction. The probability 
that an event has a particular condition entirely depends 
upon the probability that if the condition existed the 
event would follow. If we 'take up a pack of common 
playing cards, and observe that they are arranged in per- 
fect numerical order, we conclude beyond all reasonable 
doubt that they have been thus intentionally arranged 
by some person acquainted with the usual order of 
sequence. This conclusion is quite irresistible, and rightly 





80 ; for there are but two suppositions which we can make 
as to the reason of the cards being in that particular 
order : — ^ 

1. They may have been intentionally arranged by some 
one who would probably prefer the numerical order. 

2. Ihey may have fallen into that order by chance, that 
is, by some series of conditions which, being unknown to 
us cannot be known to lead by preference to the particular 
order m question. 

The latter supposition is by no means absurd, for any 
one order is as likely as any other when there is no prepon. 
demting tendency. But we can readily calculate by the 
doctrine of permutations the probability that fifty-two 
objects would fall by chance into any one particular order. 
Filty-two objects can he arranged in 52 x 51 x . . x 3 
X 2 X I or about 8066 x (io)«* possible orders, the 
number obtained requiring 6S plaoes of figures for its 
lull expression. Hence it is excessively unlikely that 
anyone should ever meet with a pack of cards arrancred 
in perfect order by accident. If we do meet with a 
pack so arranged, we inevitably adopt the other supposi- 
tion, that some person, haWng reasons for preferring that 
special order, has thus put them together. 

We know that of the immense number of possible 
orders the numerical order is the most remarkable ; it is 
useful as proving the perfect constitution of the paok, and 
It is the intentional result of certain games. At any rate 
the probability that intention should produce that order is 
incompai-ably greater than the probability that chance 
should produce it ; and as a certain pack exists in that 
order, we rightly prefer the supposition which most pro- 
oably leads to the observed result 

By a similar mode of reasoning we every day arrive 
and validly arrive, at conclusions approximating to cer- 
tainty. Whenever we observe a perfect resemblance 
between two objects, as, for instance, two printed pages 
two engravings, two coins, two foot-prints, we are war- 
i^nted m asserting that they proceed from the same type 
the same plat«, the same pair of dies, or the same boot' 
Ana why ? Because it is almost impossible that with 
amerent types, plates, dies, or boots some apparent dis- 
tinction of form should not be produced. It is impossible 

^^mm V 








for the hand of the most skilful artist to make two objecte 
alike, so that mechanical repetition is the only probable 
explanation of exact similarity. 

We can often establish with extreme probability that 
one document is copied from another. Suppose that each 
document contains io,cxx> words, and that tlie same word 
is incorrectly spelt in each. There is then a probability of 
less than i in io,cx» that the same mistake should be 
made in each. If we meet with a second error occurring 
in each document, the probability is less than i in 10,000 
X 9999' that two such coincidences should occur by chance, 
and the numbers grow with extreme rapidity for more 
numerous coincidences. We cannot make any precise 
calculations without taking into account the character of 
the errors committed, concerning the conditions of which 
we have no accurate means of estimating probabilities. 
Nevertheless, abundant evidence may thus be obtained 
as to the derivation of documents from each other. In 
the examination of many sets of logarithmic tables, six 
remarkable errors were found to be present in all but 
two, and it was proved that tables printed at Paris, Berlin, 
Florence, Avignon, and even in China, besides thirteen 
sets printed in England between the years 1633 and 1822, 
were derived directly or indirectly from some common 
source.^ With a certain amount of labour, it is possible 
to establish beyond reasonable doubt the relationship or 
genealogy of any number of copies of one document, pro- 
ceeding possibly from parent copies now lost The rela- 
tions between the manuscripts of the New Testament have 
been elaborately investigated in this manner, and the same 
work has been performed for many classical writings, 
especially by German scholars. 

Principle of the Inverse Method, 

The inverse application of the rules of probability 
entirely depends upon a proposition which may be thus 
stated, nearly in the words of Laplace.* If an event can 

^ Tiardner, Edinburgh Review, July 1834, p. 277. 
■ Mimoires par divert Savons, torn, vl ; quoted by Todhunter in 
his Hiitory of the Theory of FrobabHity, p. 458. 


he produced by any one of a ceHain number of different 
catise^, all eqmlly probable a priori, the probabilities of the 
existence of tliese causes as infeired from the event, are pro- 
portional to tlie probabilities of the event as derived from these 
causes. In other words, the most probable cause of an 
event which has happened is that which would most pro- 
bably lead to the event supposing the cause to exist; but 
all other possible causes are also to be taken into account 
with probabilities proportional to the probability that the 
event would happen if the cause existed. Suppose, to fix 
our ideas clearly, that E is the event, and C, Cj C3 are the 
three only conceivable causes. If C exist, the probability 
is pi that E would follow ; if Cj or Cj exist, the like pro- 
babilities are respectively p^ and p^ Then as ;?j is to p^, 
so is the probability of Cj being the actual cause to the 
probabQity of 0, being it ; and, similarly, as p^ is to p., so 
13 the probability of C, being the actual cause to the 
probability of Cj being it By a simple mathematical pro- 
cess we arrive at the conclusion that the ** ^ual probability 
of Cj being the cause is 

Pi + Pt + Pi* 
and the similar probabilities of the existence of C, and 
C3 are, * 

, ^« and ^ 

Pi-tPt-hPi Pi+Pi+Pi 

The sum of these three fractions amounts to unity, which 

correctly expresses the certainty that one cause or other 

must be in operation. 

We may thus state the result in general language. 
If it is certain that one or other of the supposed cames 
exists, the probability that any one does exist is the proba- 
bility that if it exists the event happens, divided by the mm 
of all the similar probabilities. Tlierfe may seem to be an 
mtncacy in this subject which may prove distasteful to 
some readers ; but this intricacy is essential to the subject 
in hand. No one can possibly understand the principles 
of inductive reasoning, unless he wiU take the trouble to 
master the meaning of this rule, by which we recede from 
an event to the probability of each of its possible causes. 

This rule or principle of the indirect method is that 
which common sense leads us to adopt almost instinctively, 

R 2 


• ( 




before we have any comprehension of the principle in its 
general form. It is easy to see, too, that it is the rule 
'which will, out of a great multitude of cases, lead us most 
often to the truth, since the most probable cause of an 
event really means that cause which in the greatest 
number of cases produces the event Donkin and Boole 
have given demonstrations of this principle, but the one 
most easy to comprehend is that of Poisson. He imagines 
each possible cause of an event to be represented by a 
distinct ballot-box, containing black jind white balls, in 
such a ratio that the probability of a white ball being 
drawn is equal to that of the event happening. He further 
supposes that each box, as is possible, contains the same 
total number of balls, black and white ; then, mixing all 
the contents of the boxes together, he shows that if a 
white ball be drawn from the aggregate ballot-box thus 
formed, the probability that it proceeded from any par- 
ticular ballot-box is represented by the number of white 
balls in that particular box, divided by the total number 
of white balls in all the boxes. This result corresponds to 
that given by the principle in question.^ 

Thus, if there be three boxes, each containing ten balls 
in all, and respectively containing seven, four, and three 
white balls, then on mixing all the balls together we have 
fourteen white ones ; and if we draw a white ball, that is 
if the event happens, the probability that it came out of 


the first box is J^ ; which is exactly equal to , . V t" s ' 

Tff + TTF + TTF 

the fmction given by the rule of the Inverse Method. 

Simple Applications of the Inverse Method. 

In many cases of scientific induction we may apply the 
principle of the inverse method in a simple manner. If 
only two, or at the most a few hypotheses, may be made 
as to the origin of certain phenomena, we may sometimes 
easily calculate the respective probabilities. It was thus 
that Bunsen and Kirchlioff established, with a probability 
ittle short of certainty, that iron exists in the sun. On 
comparing the spectra of sunlight and of the light proceed- 

I Poiason, lUeherchu iur la ProbabilUe da JugemcuU, Paiia, 1837, 
W^ 82, 83. 


ing from the incandescent vapour of iron, it became appa- 
rent that at least sixty bright lines in the spectrum of iron 
coincided with dark lines in the sun's spectrum. Such coin- 
cidences could never be observed with certainty, because, 
even if the lines only closely approached, the instrumental 
imperfections of the spectroscope would make them appa- 
rently coincident, and if one line came within half a milli- 
metre of another, on the map of the spectra, they could not 
be pronounced distinct. Now the average distance of the 
solar Imes on Kirchhofif's map is 2 mm., and if we throw 
down a line, as it were, by pure chance on such a map, 
the probability is about one-half that the new line will fall 
within J mm. on one side or the other of some one of the 
solar lines. To put it in another way, we may suppose 
that each solar line, either on account of its real breadth, 
or the defects of the instrument, possesses a breadth of 
i mm., and that each line in the iron spectrum has a like 
breadth. The probability then is just one-half that the 
centre of each iron line will come by chance within i mm. 
of the centre of a solar line, so as to appear to coincide 
with it The probability of casual coincidence of each 
iron line with a solar line is in like manner i. Coinci- 
dence in the case of each of the sixty iron lines is a very 
unlikely event if it arises casually, for it would have a 
probability of only {^)^ or less than i in a trHlion. The 
odds, in short, are more than a million million millions 
to umty against such casual coincidence.^ But on the 
other hypothesis, that iron exists in the sun, it is highly 
probable that such coincidences would be observed ; it is 
immensely more probable that sixty coincidences would be 
observed if iron existed in the sun, than that they .should 
arise from chance. Hence by our principle it is immensely 
probable that iron does exist in the sun, 

AH the other interesting results, given by the comparison 
of spectra, rest upon the same principle of probability. 
Ihe almost complete coincidence between the spectra of 
Bolar, lunar, and planetary light renders it pracMcally 
certain that the light is all of solar origin, and is reflected 
trom the surfaces of the moon and planets, suffering onh 

1 ! ,^»rchhoff's Researches <m the Solar Spectrum. 
M»tod by Roanoe, pp. 18, 19. 

Fiist part, trans- 






Blight alteration from the atmospheres of some of tlie 
planets. A fresh confirmation of the truth of the Coper- 
nican theory is thus furnished. 

Herschel proved in this way the connection between the 
direction of the oblique faces of quartz crystals, and 
the direction in which the same crystals rotate the 
plane of polarisation of light. For if it is found in a 
second crystal that tlie relation is the same as in the first, 
the probability of this happening by chance is J ; the 
probability that in another crystal also the direction 
will be the same is i, and so on. The probability that 
in n 4- I crystals there would be casual agi-eement of direc- 
tion is the nth power of i. Thus, if in examining fourteen 
crystals the same relation of the two phenomena is dis- 
covered in each, the odds that it proceeds from uniform 
conditions are more than 8000 to i.^ Since the first 
observations on this subject were made in 1820, no excep- 
tions have been observed, so that the probability of in- 
variable connection is incalculably great. 

It is exceedingly probable that the ancient Egyptians 
had exactly recorded the eclipses occurring during long 
periods of time, for Diogenes Laertius mentions that 373 
solar and 832 lunar eclipses had been observed, and the 
ratio between these numbers exactly expresses that which 
would hold true of the eclipses of any long period, of 
say 1200 or 1300 years, as estimated on astronomical 
grounds. It is evident that an agreement between small 
numbers, or customary numbers, such as seven, one 
hundred, a myriad, &c., is much more likely to happen from 
chance, and therefore gives much less presumption of de- 
pendence. If two ancient writers spoke of the sacrifice of 
oxen, they would in all probability describe it as a heca* 
tomb, and there would be nothing remarkable in the coin- 
cidence. But it is impossible i;o point out any special 
reason why an old writer should select such numbers as 
373 and 832, unless they had been the results of observa- 

On similar grounds, we must inevitably believe in the 

' Edinburgh Review^ No. 185, vol. xcii. July 1850, p. 32 ; Herschel's 
, p. 421 ; Transixctioii* of the Cambridge I'hilosophical iiodcty, 

vuJ. i. JI.43. 


human origin of the flint flakes so copiously discovered of 
late years. For though the accidental stroke of one stone 
against another may often produce flakes, such as are 
occasionally found on the sea-shore, yet when several 
flakes are found in close company, and each one bears 
evidence, not of a single blow only, but of several suc- 
cessive blows, all conducing to form a symmetrical knife- 
like form, the probability of a natural and accidental 
origin becomes incredibly small, and the contrary suppo- 
sition, that they are the work of intelligent beings, 
approximately certain.^ 

The TJieory of Prohahility in Astronomy, 

The science of astronomy, occupied with the simple 
relations of distance, magnitude, and motion of the 
heavenly bodies, admits more easily than almost any 
other science of interesting conclusions founded on the 
theory of probability. More than a century ago, in 
1767, Michell showed the extreme probability of bonds 
connecting together systems of stai-s. He was struck 
by the unexpected number of fixed stars which have 
companions close to them. Such a conjunction mi^^ht 
happen casually by one star, although possibly at' a 
great distance from the other, happening to Lie on a 
straight line passing near the earth. But the probabilities 
are so greatly against such an optical union happening 
often in the expanse of the heavens, that Michell asserted 
the existence of some connection between most of the 
double stai's. It has since been estimated by Struve, 
that the odds are 9570 to i against any two stars of not 
less than the seventh magnitude falling within the appa- 
rent distance of four seconds of each other by chance, and 
yet ninety-one such cases were known when the estimation 
was made, and many more cases have since been discovered. 
There were also four known triple stars, and yet the odds 
against the appearance of any one such conjunction are 
' 73*524 to I.* The conclusions of Michell have been 

* Evans* Ancient Stone Implementt of Great Britain. London, 
1872 (Longmans). 

^Herschel, Outliius of Astronomy, 1849, p. 565 ; but ToJlhunter, 
in his Hittory of the Theory of Probability, p. 335, states that the 
calculations do not agree with those published by Struve. 

I 1} 





entirely verified by the discovery that many double stars 
are connected by gravitation. 

Michell also investigated the probability that the six 
brightest stars in the Pleiades should have come by 
accidents into such striking proximity. Estimating the 
number of stars of equal or greater brightness at 1500, he 
found the odds to be nearly 500,000 to i against casual 
conjunction. Extending the same kind of argument to 
other clusters, such as that of Pi-msepe, the nebula in the 
hilt of Perseus* sword, he says:^ "We may with the 
highest probability conclude, tho odds against the contrary 
opinion being many million millions to one, that the stars 
are really collected together in clusters in some places, 
where they form a kind of system, while in others there 
are either few or none of them, to whatever cause this may 
!!>e owing, whether to their mutual gravitation, or to some 
other law or appointment of the Creator." 

The calculations of Michell have been called in question 
by the late James D. Forbes,^ and ^Ir. Todhunter vaguely 
countenances his objections,' otherwise I should not have 
thought them of much weight. Certainly Laplace accepts 
Michell's views,* and if Michell be in error it is in the 
methods of calculation, not in the general validity of his 
reasoning and conclusions. 

Similar calculations might no doubt be applied to the 
peculiar drifting motions which have been detected by 
Mr. R A. Proctor in some of the constellations.* The odds 
are veiy greatly against any numerous group of stars mov- 
ing together in any one direction by chance. On like 
grounds, there can be no doubt that the sun has a con- 
siderable proper motion because on the average the fixed 
btars show a tendency to move apparently from one point 
of the heavens towards that diametrically opposite. The 
sun's motion in the contrary direction would explain this 
tendency, otherwise we must believe that thousands of 
stars accidentally agree in their direction of motion, or are 

* Philoiophical TrantactionSf 1767, vol Ivii p. 431. 
' PhilosophiccU Magazinty 3rd Senee, voL xxxvii. p. 401, December 
i8qo ; also August 1849. 

^trtory, &c., p. 334. * Euai FhUosophique, p. 57. 

Proceedings of the Royal Society y 20 January, 1870 ; Philosophical 
aiagazine^ 4th Series, vol. xxxix. p. 381. 



urged by some common force from which the sun is 
exempt. It may be said that the rotation of the earth is 
proved in like manner, because it is immensely more pro- 
bable that one body would revolve than that the sun 
moon, planets, comets, and the whole of the stars of the 
heavens should be whiried round the earth daily, with a 
uniform motion superadded to their own peculiar motions. 
This appears to be mainly the reason which led Gilbert 
one of the eariiest English Copemicans. and in every way 
an admirable physicist, to admit the rotation of the earth 
while Francis Bacon denied it 

In contemplating the planetary system, we are struck 
with the similarity in direction of nearly all its movements 
Newton remarked upon the regularity and uniformity of 
these motions, and contrasted them with the eccentricity 
and irregularity of the cometary orbits.^ Could we in 
fact, look down upon the system from the northern side 
we should see all the planets moving round from west to 
east, the satellites moving round their primaries, and the 
sun planets, and satellites rotating in the same direction, 
with some exceptions on the verge of the system. In the 
time of Laplace eleven planets were known, and the direc- 
tions of rotation were known for the sun, six planets the 
satellites of Jupiter, Saturn's ring, and one of his satellites 
Ihus there were altogether 43 motions all concurrin^r 
namely : — ^* 

Orbital motions of eleven planets . . 1 1 
Orbital motions of eighteen satellites . .18 
Axial rotations ! 14. 


The probabiHty that 43 motions independent of each 

other would coincide by chance is the 42nd power of i, so 
that the odds are about 4,400.000,000,000 to i in favour of 
some common cause for the uniformity of direction. This 
probability, as Laplace obsei-ves,2 is higher than that of 
many historical events which we undoubtingly believe In 
the present day, the probability is much increased by the 
discovery of additional planets, and the rotation of other 

! -Pnnctpta, bk. ii. General scholium- 
S,tu^"l* ^^*Vo«op;it^, p. 55. Laplace appears to count the rings of 
S'ltum as giving two independent movemeiite. ^ 







satellites, and it is only slightly weakened by the fact that 
some of the outlying satellites are exceptional in direction, 
there being considerable evidence of an accidental dis- 
turbance in the more distant parts of the system. 

Hardly less remarkable than the uniform direction of 
motion is the near approximation of the orbits of the 
planets to a common plane. Daniel Bernoulli roughly 
estimated the probability of such an agreement arising 
from accident as l -5- (12)® the greatest inclination of any 
orbit to the sun's equator being I-I2th part of a quadrant. 
Laplace devoted to this subject some of his most ingenious 
investigations. He found the probability that the sum of 
the inclinations of the planetary orbits would not exceed 
by accident the actual amount (•914187 of a right angle' 
for the ten planets known in 1801) to be (^^y (9 14 187),*** 

or about •00000011235. This probability may be com- 
bined with that derived from the direction of motion, and 
it then becomes immensely probable that the constitution 
of the planetary system arose out of uniform conditions, 
or, as we say, from some common cause.^ 

If the same kind of calculation be applied to the orbits 
of comets, the result is very different.' Of the orbits 
which have been determined 48*9 per cent, only are direct 
or in the same direction as the planetary motions.* Hence 
it becomes apparent that comets do not properly belong 
to the solar system, and it is probable that they are stray 
portions of nebulous matter which have accidentally become 
attached to the system by the attractive powers* of the 
sun or Jupiter. 

The General Inverse Problem, 

In the instances described in the preceding sections, 
we have been occupied in receding from the occurrence 
of certain similar events to the probability that there 

> Lubbock, Essay on Frobability, p. 14. De Morgan, Encye. 
Metrap. art. Probability ^ p. 412. Tod hunter's History of the Theory 
of Probabilityj p. 543. Concerning the objections raised to these 
conclusions by Boole, see the Philosophical Magazine, 4tb Series, 
vol. ii. p. 98. Boole's Latos of Thought^ pp. 364-375, 

2 Laplace, Essai Philosophiaue^ pp. 55, 56. 

> Chambers* Asironomy, 2nd ed. pp. 346-40, 



I i 


must have been a condition or cause for such events. We 
have found that the theory of probability, although never 
yielding a certain result, often enables us to establish an 
hypothesis beyond the reach of reasonable doubt. There 
is, however, another method of applying the theory, 
which possesses for us even greater interest, because it 
illustrates, in the most complete manner, the theory of 
inference adopted in this work, which theory indeed it 
suggested. The problem to be solved is as follows : — 

An event Jtaving liappened a certain mimher of times, 
and failed a certain number of times, required the pro- 
lability tJuit it wUl happen any given number of times 
in the future under the same circumstances. 

AH the larger planets hitherto discovered move in one 
direction round the sun ; what is the probability that, if a 
new planet exterior to Neptune be discovered, it will move 
in the same direction ? All known permanent gases, ex- 
cept chlorine, are colourless ; what is the probability that, 
if some new permanent gas should be discovered, it will 
be colourless ? In the general solution of this problem, we 
wish to infer the future happening of any event from' the 
number of times that it has already been observ^ed to 
happen. Now, it is very instructive to find that there is 
no known process by which we can pass directly from the 
data to the conclusion. It is always requisite to recede 
from the data to the probability of some hypothesis, and 
to make that hypothesis the ground of our inference 
concerning future events. Mathematicians, in fact, make 
every hypothesis which is applicable to the question in 
hand ; they then calculate, by the inverse method, the 
probability of every such hypothesis according to the 
data, and the probability that if each hypothesis be true, 
the required future event will happen. The total pro- 
bability that the event will happen is the sum of the 
separate probabilities contributed by each distinct hypo- 

To illustrate more precisely the method of solving the 
problem, it is desii-able to adopt some concrete mode of 
representation, and the ballot-box, so often employed by 
mathematicians, will best serve our purposa Let the 
happening of any event be represented by the drawing of 
a white tall from a ballot-box, while the fiedlure of an 


-, M 





[on A p. 

event is represented by the drawing of a black ball. Now, 
in the inductive problem we are supposed to bo ignorant 
of the contents of the ballot-box, and are required to 
ground all our inferences on our experience of those con- 
tents as shown in successive drawings. Rude common 
sense would guide us nearly to a tnie conclusion. Thus, 
if we had drawn twenty balls one after another, replacing 
the ball after each drawing, and the ball had in each case 
proved to be white, we should believe that there was a 
considerable preponderance of white balls in the urn, and 
a probability in favour of drawing a white ball on the next 
occasion. Though we had drawn white balls for 
thousands of times without fail, it would still be possible 
that some black balls lurked in the urn and would at last 
appear, so that our inferences could never be certain. On 
the other hand, if black balls came at intervals, we should 
expect that after a certain number of trials the black balls 
would appear again from time to time with somewhat the 
same frequency. 

The mathematical solution of the question consists in 
little more than a close analysis of the mode in which our 
common sense proceeds. If twenty white balls have been 
drawn and no black ball, my common sense tells me that 
any hypothesis which makes the black balls in the urn 
considerable compared with the white ones is improbable ; 
a preponderance of white balls is a more probable hypo- 
thesis, and as a deduction from this more probable hypo- 
thesis, I expect a recurrence of white balls. The mathe- 
matician merely reduces this process of thought to exact 
numbers. Taking, for instance, the hypothesis that there 
are 99 white and one black ball in the urn, he can calcu- 
late the probability that 20 white balls would be drawn 
in succession in those circumstances; he thus forms a 
definite estimate of the probability of this hypothesis, and 
knowing at the same time the probability of a white ball 
reappearing if such be the contents of the urn, he com- 
bines these probabilities, and obtains an exact estimate 
that a white hall will recur in consequence of this hypo- 
thesis. But as this hypothesis is only one out of many 
possible ones, since the ratio of white and black balls may 
be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has to 
repeat the estimate for every such possible hypothesis. 


To make the method of solving the problem perfectly 
evident, I will describe in the next section a very simple 
case of the problem, originally devised for the purpose by 
Condorcet, which was also adopted by Lacroix,i and has 
passed into the works of De Morgan, Lubbock, and others. 

Simple IlludrcUion of the Inverse Problem, 

Suppose it to be known that a ballot-box contains only 
four black or white balls, the ratio of black and white balls 
being unknown. Four drawings having been made with 
replacement, and a white ball having appeared on each 
occasion but one, it is required to determine the proba- 
bility that a white ball will appear next time. Now the 
hypotheses which can be made as to the contents of the 
urn are very limited in number, and are at most the 
following five : — 

4 white and o black balls 



















^ »» » 4 » » 

The actual occurrence of black and white balls in the 
drawings puts the first and last hypothesis out of the 
question, so that we have only three left to consider. 

If the box contains three white and one black, the 
probability of drawing a white each time is }, and a black 
i ; so that the compound event observed, namely, three 
white and one black, has the probability J X } X | x J, by 
the rule already giveu (p. 204). But as it is indifferent 
in what order the balls are drawn, and the black ball 
might come first, second, third, or fourth, we must multi- 
ply by four, to obtain the probability of three white and 
one black in any order, thus getting JJ. 

Taking the next hypothesis of two white and two 
black balls in the urn, we obtain for the same proba- 
bility the quantity J x J x J x J x 4, or ^J, and from the 
thmi hypothesis of one white and three black we deduce 
likewise i x i x J x J x 4, or ^. According, then, as we 

* Traite iUnuntaire du Caleul det ProbabiliUi, 3rd ed. 08^^^ 
!>. 148. ^ ^^^* 


< ; M 








adopt the first, second, or third hypothesis, the proba- 
bility that the result actually noticed would follow is ||, 
J4, and ^^. Now it is certain that one or other of these 
hypotheses must be the true one, and their absolute 
probabilities are proportional to the probabilities that the 
observed events would follow from them (pp. 242, 243). All 
we have to do, then, in order to obtain the absolute pro- 
bability of each hypothesis, is to alter these fractions in 
a uniform ratio, so that their sum shall be unity, the 
expression of certainty. Now, since 27 + 16 + 3 = 46, 
this will be effected by dividing each fraction by 46, and 
multiplying by 64. Thus the probabilities of the first, 
second, and third hypotheses are respectively — 

27 16 3 

46' 46* 46' 

The inductive part of the problem is completed, since we 
have found that the urn most likely contains three white 
and one black ball, and have assigned the exact probability 
of each possible supposition. But we are now in a position 
to resume deductive reasoning, and infer the probability 
that the next drawing will yield, say a white ball. For if 
the box contains three white and one black ball, the pro- 
bability of drawing a white one is certainly J ; and as the 
probability of the box being so constituted is JJ, the com- 
pound probability that the box will be so filled and will 
give a white ball at the next trial, is 

27 3 81 
-^ X 7 or - . 



Again, the probability is jj that the box contains two 
white and two black, and under those conditions the 
probability is J that a white ball will appear ; hence the 
probability that a white ball will appear in consequence 
of that condition, is 

16 ^ I 32 

56 ^ 5 **' 184* 

From the third supposition we get in like manner the 

Since one and not more than one hypothesis can be true. 


we may add together these separate probabilities, and we 
find that 

«! , 32 , 3 116 

184 "^ 184 ■*■ i"8i *''' T8i 
IS the complete probability that a white ball will be next 
drawn under the conditions and data supposed. 

Gemral SohUion of the Inverse Problem, 

In the instance of the inverse method described in the 
last section, the balls supposed to be in the ballot-box 
were few, for the purpose of simplifying the calculation. 
m order that our solution may apply to natural phe- 
nomena we must render our hypotheses as little arbitrary 
as possible. Having no d priori knowledge of the con- 
ditions of the phenomena in question, there is no limit 
to the variety of hypotheses which might be suggested. 
Mathematicians have therefore had recourse to the most 
extensive suppositions which can be made, namely, that 
the ballot-box contains an infinite number of balls- they 
have then varied the proportion of white to black balls 
continuously, froni the smallest to the greatest possible 
proportion, and estimated the aggregate probability which 
results from this comprehensive supposition. 

To explain their procedure, let us imagine that, instead 
of an infinite number, the ballot-box contains a large 
finite number of balls, say 1000. Then the number of 
white balls might be I or 2 or 3 or 4, and so on, up to 
999. Supposing that three white and one black ball 
have been drawn from the urn as before, there is a certain 
very small probability that this would have occurred in 
the case of a box containing one white and 990 black 
balls ; there is also a smaU probability that from such a 

^\ l-f-..^®'^* Y^ ^^^^^ ^ w^i^- Compound these 
probabdities, and we have the probability that the next 
ball really will bo white, in consequence of the existence 
of that proportion of baUs. If there be two white and ogS 
black balls m the box, the probabOity is greater and will 
increase until the balls are supposed to be in the proper- 
tion of tho^ drawn. Now 999 different hypotheses are 
possible, and the calculation is to be made for each of 
tnese, and their aggregate taken as the final result. It is 





apparent that as the number of balls in the box is increased, 
the absolute probability of any one hypothesis concerning 
the exact proportion of balls is decreased, but the aggregate 
results of all the hypotheses will assume the character of 

a wider average. - x,- 

When we take the step of supposing the balls withm 
the urn to be infinite in number, the possible proportions 
of white and black balls also become infinite, and the 
probability of any one proportion actually existing is 
infinitely small. Hence the final result that the next ball 
drawn will be white is really the sum of an infinite 
number of infinitely small quantities. It might seem 
impossible to calculate out a problem having an infinite 
number of hypotheses, but the wonderful resources of the 
integral calculus enable this to be done with far greater 
facility than if we supposed any large finite number of 
balls, and then actually computed the results. I will not 
attemp.t to describe the processes by which Laplace finally 
accomplished the complete solution of the problem. They 
are to be found described in several English works, espe- 
cially De Morgan's Treatise on Probabilities, in the Encij- 
dopcedia Metropolitana, and Mr. Todhunter's History of 
the Tluory of Probability. The abbreviating power of 
mathematical analysis was never more strikingly shown. 
But I may add that though the integral calculus is 
employed as a means of summing infinitely numerous 
results, we in no way abandon the principles of com- 
binations already treated. We calculate the values of 
infinitely numerous factorials, not, however, obtaining 
their actual products, which would lead to an infinite 
number of figures, but obtaining the final answer to the 
problem by devices which can only be comprehended 
after study of the integiul calculus. 

It must be allowed that the hypothesis adopted by 
Laplace is in some degree arbitrary, so that there was some 
opening for the doubt which Boole has cast upon it.^ 
But it° may be replied, (i) tliat the supposition of an 
infinite number of balls treated in the manner of Laplace 
is less arbitrary and more comprehensive than any other 
that can be suggested. (2) The result does not differ 

> Law of Thought, pp. 368-375« 



much from that which would be obtained on the hypothesis 
of any large finite number of balls. (3) The supposition 
leads to a series of simple formulas which can be applied 
with ea^e m many cases, and which bear aU the appearance 
ot truth so far as it can be independently judged by a 
sound and practiced understanding. 

Rules of the Inverse Method, 

By the solution of the problem, as described in the last 
section, we obtain the following series of simple rules 

I. To find th^ probabUity that an event which has not 
hjiherto been observed to fail will happen once more, 
dimde the nurriber of times the event has been observed 
increased by one, bi the same number increased by two 

If there have been m occasions on which a certain event 
might have been observed to happen, and it has happened 
on all those occasions, then the probabiHty that it will 
happen on the next occasion of the same kind is ^^LdLl. 
For instance, we may say that there are nine pla'^e^ in 
tlie planetary system where planets might exist obeying 
Bodes law of distance, and in every place there is a 
planet obeying the law more or less exactly, althouch 
no reason is known for the coincidence. Hence the 
probability that the next planet beyond Neptune will 
conform to the law is |f . 

2. To find the probaklUy that an event which has not 
hUherto failed will not fail for a certain number of new 
occasions, divide the number of times the event has hav^ 
V^ increased by one, by th^ same number increased bv 
»neand the nurriber of times it is to happen. 

An event having happened m times without fail, the 
probabiHty that it will happen n more times is **+' 
^us the probability that three new planets w^iJd^obey 
^de s law is « ; but it must be aUowed that this, as weU 
for fL?TT ^^lt>,wo^ld be much weakened by the 
fact that Neptune can barely be said to obey the law 

numht^^T''^ I^^"^. ^PP^^ «^ A^ a certain 
ICZff'^'J''^'^.^^ ^o6a^% that ii will happen 
W(5 r^ tim^, divide the n.uwher of times the eoe^ithas 










hajrpened increased hy one, hy the whole numher of times 
the event has happened or failed in>creased hy tvx). 

If an event has happened m times and failed n times, 
the probability that it will happen on the next occasion ia 

— !!!Li-L_. Thus, if we assume that of the elements dis- 

covered up to the year 1873, 50 are metallic and 14 non- 
metallic, then the probability that the next element dis- 
covered will be metallic is ^. Again, since of 37 metals 
which have been sufficiently examined only four, namely, 
sodium, potassium, lanthanum, and lithium, are of less 
density than water, the probability that the next metal 
examined or discovered wiU be less dense than water is 

--^— j- — or -* . 
37 + 2 39 

We may state the results of the method in a more 
general manner thus,^ — If under given circumstances cer- 
tain events A, B, C, &c., have happened respectively m, n, 
p, &c., times, and one or other of these events must 
happen, then the probabilities of these events are propor- 
tional to m + I, n + I, i> + I, Ac, so that the probability 

of A will be 

m-\- I 

But if new 


events may happen in addition to those which have been 
observed, we must assign unity for the probability of such 
new event. The odds then become i for a new event, 
m + I for A, n + I for B, and so on, and the absolute 

probability of A is — ; , **, — ; r-5— • 

It is interesting to trace out the variations of probability 
according to these rules. The first time a casual event 
happens it is 2 to i that it will happen again ; if it does 
happen it is 3 to i that it will happen a third time ; and 
on successive occasions of the like kind the odds become 
4, 5^ 6, &c., to I. The odds of course will be discriminated 
from the probabilities which are successively }, }, |, &c. 
Thus on the first occasion on which a person sees a shark, 
and notices that it is accompanied by a little pilot fish, 
the odds are 2 to i, or the probability }, that the next 
shark will be so accompanied. 

* De Morgan's Eisaiy on Probabilities, Cabinet CycIoptediA. p. 67. 

When an event has happened a very great number of 
times. Its happenmg once again approaches nearly to cer- 
^!S^y- .. ^® suppose the sun to have risen one thousand 
million times, the probability that it wiU rise again, on 
the ground of this knowledge merely, is '»Qoo,ooo,ooq 4- i 

But then the probability that it will conSZTistfotL 
long a period in the future is only '>«)o,ooo,ooo + 1 ^^ ^^^^ 

exactly i The probabiHty that itQ'^Se'so rising a 
thousand times as long is only about ^^. The lesson which 
we may draw from these figures is'^te that which we 

n^v^ Jfj Z""^^? ^?^^'' ^^"^^ly^ ^^^^ experience 
Tr^Zlf^w w"^ knowledge, and that it is exceedingly 
mprobable that events mil always happen as we observe 
them. Inferences pushed far beyond their data soon lose 
any considerable probability. De Morgan has 8aid,i " No 
fimte experience whatsoever can justify us in saying that 
nr fW I^ '^"^ comcide with the past'inall time to come, 
or that there is any probabihty for such a conclusion." On 
tiie other hand, we gain the assurance that experience 

k^^!. ^ t^^f'^ ^^^ P'^^^^^^ ^ill give ^is the 
n^Sf ^^^?;^^v«"t« with an unUmited degree of 

subject to arbitrary interference 

onfv Zh\^ "-"^"l^ understood that these probabilities are 
only such as arise from the mere happening of the events 

3^W%f '"^ ^T^^'^^ ^^^^^' ^-- other%Turts 
concerning those evente or the general laws of nature. 

AU our knowledge of nature is indeed founded in like 
manner upon observation, and is therefore only probable. 
The law of gravitation itself is only probably true BiS 
when a number of different facts, observed under ?he most 
— circumstances, are found to be harmonized under a 
supposed law of nature, the probability of the law approxi- 
mates dosely to certainty. ^Each science rests upoHo 

31^""^'^ ^^^;. *°^ ^"^^^ «^ °^^<^^ ««PP«rt from 
nnalogies or connections with other sciences, that ther^ 

D^b^nr r'^^ ^''^. T^^ ^^^^ ^^^ judgment of the 
probability of an event depends entirely upoi a few ante- 

» EMtay on Probabilitiet, p. 128. 

s 2 



U I 



cedent events, disconnected from the general body of 
physical science. 

Events, again, may often exhibit a regularity of suc- 
cession or preponderance of character, which the simple 
formula will not take into account. For instance, the 
majority of the elements recently discovered are metals, 
so that the probability of the next discovery being that of 
. a metal, is doubtless greater than we calculated (p. 258). 
At the more distant parts of the planetary system, there 
are symptoms of disturbance which woidd prevent our 
placing much reliance on any inference from the prevailing 
order of the known planets to those undiscovered ones 
which may possibly exist at great distances. These and 
all like complications in no way invalidate the theoretic 
truth of the fonnuhis, but render their sound application 
much more difficult. 

Erroneous objections have been raised to the theory of 
probability, on the ground that we ought not to trust to 
our d priori conceptions of what is likely to happen, but 
should always endeavour to obtain precise experimental 
data to guide us.^ This course, however, is perfectly in 
accordance with the theory, which is our best and only 
guide, whatever data we possess. We ought to be always 
applying the inverse method of probabilities so as to take 
into account all additional information. When we throw 
up a coin for the first time, we are probably quite ignorant 
whether it tends more to fall head or tail upwards, and 
we must therefore assume the probability of each event 
as ^. But if it shows head in the firat throw, we now 
have very slight experimental evidence in favour of a 
tendency to show head. The chance of two heads is 
now slightly greater than J, which it appeared to be at 
first,* and as we go on throwing the coin time after time, 
the probability of head appearing next time constantly 
varies in a slight degree according to the character of our 
previous experienca As Laplace remarks, we ought 
always to have regard to such considerations in common 
life. Events when closely scrutinized will hardly ever 
prove to be quite independent, and the slightest pre- 

* J. S. Mill, System of Logte, 5th edition, bk. iii. chap, xviii. § 3. 
^ Todliuuter's Uiitoryy pp. 472, 598 



ponderance one way or the other is some evidence of 
connection, and m the absence of better evidence should 
be taken into account. 

The grand object of seeking to estimate the probabUity 
of future events from past experience, seems to have beon 
entertained by James Bernoulli and De Moivre, at least 
such w^ the opinion of Condorcet ; and BernouUi may be 
said to have solved one case of the problem.^ The English 
wntei-s Bayes and Price are, however, undoubtedly the 
tet who put forward any distinct rules on the subject 2 
Condorcet and several other eminent mathematicians ad- 
vanced the mathematical theory of the subject : but it was 
reserved to tlie immortal Laplace to bring to the subject 
the fuU power of Iils genius, and carry the solution of the 
problem almost t« perfectioa It is instructive to observe 
that a theoiy which arose from petty games of chance, the 
rules and the very names of which are forgotten, m-ad nail v 
advanced, until it embraced the most sublime problems of 
science and finally undertook to measure the value and 
certainty of all our inductions. 

Fortuitous Coincidences, 

We should have studied the theory of probability to 
very little purpose, if we thought that it would furnish 
us with an infallible guide. The theory itself points 
out the approximate certainty, that we shall sometimes 
De deceived by extraordinary fortuitous coincidences, 
ihere 13 no run of luck so extreme that it may not 

wt?^"' ^""^ ^^ ""^^ ^^PP^^ *o ^' or in our time, as 
well as to other persons or in other times. We may be 
lorced by correct calculation to refer such coincidence? 
w a necessary cause, and yet we may be deceived. All 

l.ti,'.?^?'^''^ ""^ probability pretends to give, is ihs 
resuit %n the long run, as it is caUed, and this really means 

Tnw? *'V^''*^^ ?^ "^^^ ^^^g ^^y finite experience, 
tlofiy®' long, chances may be against us. Nevertheless 

thfl 7 '^ *^^ ^^* S^^^^ ^0 can have. If we always 
tmnk and act according to its well-interpreted indications, 

J paS!1^^* ?^' pp- 378. 379. 






i < .'■ 


we shall have the best chance of escaping error ; and if all 
persons, throughout all time to come, obey the theory in 
like manner, they will undoubtedly thereby reap the 
greatest advantaga 

No rule can be given for discriminating between 
coincidences which are casual and those which are the 
effects of law. By a fortuitous or casual coincidence, we 
mean an agreement between events, which nevertheless 
arise from wholly independent and different causes or con- 
ditions, and which will not always so agree. It is a 
fortuitous coincidence, if a penny thrown up repeatedly 
in various ways always falls on the same side ; but it 
would not be fortuitous if there were any similarity 
in the motions of the hand, and the height of the thi-ow, 
so as to cause or tend to cause a uniform result. Now 
among the infinitely numerous events, objects, or relations 
in the universe, it is quite likely that we shall occasionally 
notice casual coincidences. There are seven intervals in 
the octave, and there is nothing very improbable in the 
colours of the spectrum happening to be apparently 
divisible into the same or similar series of seven intervals. 
It is hardly yet decided whether this apparent coincidence, 
with which Newton was much struck, is well founded or 
not,^ but the question will probably be decided in the 

It is certainly a casual coincidence which the ancienU 
noticed between the seven vowels, the seven strings of the 
lyre, the seven Pleiades, and the seven chiefs at Thebes.' 
The accidents connected with the number seven have mis- 
led the human intellect throughout the historical period. 
Pythagoras imagined a connection between the seven 
planets and the seven intervals of the mouochord. The 
alchemists were never tired of drawing inferences from 
the coincidence in numbers of the seven planets and the 
seven metals, not to speak of the seven days of the 

A singular circumstance was pointed out concerning 
the dimensions of the earth, sun, and moon; the sun*6 
diameter was almost exactly no times as great as the 

> Newton's Opticks, Bk. I., Part il Prop. 3 ; Nature, toL l p 286 
* Axutotle's MetaphysieSf xiil 6. 3. 


earth's diameter, while in almost exactly the same ratio 
the mean distance of the earth was greater than the sun's 
diameter, and the mean distance of the moon from the 
earth was greater than the moon's diameter. The agree- 
ment was so close that it might have proved more than 
casual, but its fortuitous character is now sufficiently shown 
by the fact, that the coincidence ceases to be remarkable when 
we adopt the amended dimensions of the planetary system. 
A considerable number of the elements have atomic 
weights, which are apparently exact multiples of that 
of hydrogen. If this be not a law to be ultimately ex- 
tended to all the elements, as supposed by Prout, it is a 
most remarkable coincidence. But, as I have observed 
we have no means of absolutely discriminating accidental 
coincidences from those which imply a deep producing 
cause. A coincidence must either be very strong in 
itself, or it must be corroborated by some explanation or 
connection with other laws of nature. Little attention 
was ever given to the coincidence concerning the dimen- 
sions of the sun, earth, and moon, because it was not very 
strong in itself, and had no apparent connection with the 
principles of physical astronomy. Prout's Law bears more 
probability because it would bring the constitution of the 
elements themselves in close connection with the atomic 
theory, representing them as built up out of a simpler 

In historical and social matters, coincidences are fre- 
quently pointed out which are due to chance, although 
there is always a strong popular tendency to regard them 
as the work of design, or as having some hidden meaning. 
If to 1794, the number of the year in which Robespierre 
fell, we add the sum of its digits, the result is 181 5, the 
year in which Napoleon fell ; the repetition of the process 
gives 1830 the year in which Charles the Tenth abdicated. 
Again, the French Chamber of Deputies, in 1830, consisted 
of 402 members, of whom 221 formed the party called 
"La queue de Robespierre," while th^ .emainder, 181 in 
number, were named " Les honn^tes gens." If we give to 
each letter a numerical value corres/'-^nding to its place in 
the alphabet, it will be found that tlie sum of the values 
of the letters in each name exactly indicates the number 
of the party. 


'! ( 






A number of such coincidences, often of a very curious 
chamcter, might be adduced, and the probability against 
the occurrence of each is enormously great. They must 
be attributed to chance, because they cannot be shown 
to have the slightest connection with the general laws 
of nature ; but persons are often found to be greatly in- 
fluenced by such coincidences, regarding them as evidence 
of fatality, that is of a system of causation governing 
human afTairs independently of the ordinary laws of nature. 
Let it be remembered that there are an infinite number of 
opportunities in life for some strange coincidence to pre- 
sent itself, so that it is quite to be expected that remark- 
able conjunctions will sometimes happen. 

In all matters of judicial evidence, we must bear in 
mind the probable occurrence from time to time of un- 
accountable coincidences. The Roman jurists refused for 
this reason to invalidate a testamentary deed, the wit- 
nesses of which had sealed it with the same seal. For 
witnesses independently using their own seals might be 
found to possess identical ones by accident^ It is well 
known that circumstantial evidence of apparently over- 
whelming completeness will sometimes lead to a mistaken 
judgment, and as absolute certainty is never really attain- 
able, every court must act upon probabilities of a high 
amount, and in a certain small proportion of cases they 
must almost of necessity condemn the innocent victims 
of a remarkable conjuncture of circumstances.* Popular 
judgments usually turn upon probabilities of far less 
amount, as when the palace of Nicomedia, and even 
the bedchamber of Diocletian, having been on fire twice 
within fifteen days, the people entirely refused to believe 
that it could be the result of accident. The Romans 
believed that there was fatality connected with the name 
of Sextus. 

** Semper sub Sextis perdita Roma fuiL" 

The utmost precautions will not provide against all 
contingencies. To avoid errors in important calculations, 

1 Possiint autem omnes testes et uno annalo signare testamentum 
Qi H enim si septem anmili una sculptura fuerint, secundum quod 
Pomponio visum est ? — Justinian^ ii. tit. x. 6. 

* See Wills on CircuTnttantial Evidence n. ia8. 


it is usual to have them repeated by different computers ; 
but a case is on record in which three computers made 
exactly the same calculations of the place of a star, and 
yet all did it wrong in precisely the same manner, for no 
apparent reason.^ 

Summary of the Theory of Inductive Inference. 

Tlie theoiy of inductive inference stated in this and the 
previous chapters, was suggested by the study of the 
Inverse Method of Probability, but it also bears much 
resemblance to the so-called Deductive Method described 
by Mill, in his celebrated System of Logic. Mill's views 
concerning the Deductive Method, probably form the most 
original and valuable part of his treatise, and I should 
have ascribed the doctrine entirely to him, had I not 
found that the opinions put forward in other parts of his 
work are entirely inconsistent with the theory here upheld. 
As this subject is the most important and difficult one 
with which we have to deal, I will try to remedy the 
imperfect manner in which I have treated it, by giving a 
recapitulation of the views adopted. 

All inductive reasoning is but the inverse application 
of deductive reasoning. Being in possession of certain 
particular facts or events expressed in propositions, we 
imagme some more general proposition expressing the 
existence of a law or cause; and, deducing the particular 
results of that supposed general proposition, we observe 
whether they agree with the facts in question. Hypo- 
thesis is thus always employed, consciously or unconsci- 
ously. The sole conditions to which we need conform in 
framing any hypothesis is, that we both have and exercise 
the power of inferring deductively from the hypothesis to 
the particukr results, which are to be compared with the 
known facts. Thus there are but three steps in the process 
of induction : — 

(i) Framing some hypothesis as to the character of the 
general law. 

(2) Deducing consequences from that law. 

* MemoirioftJie EovcU Astronomical Society, vol iv. p. 200, quoted 
by Lardner, Edinburgh Review, July 1834, p. 278. 


u i ^ 










(3) Observing whether the consequences agree with the 
particular facts under consideration. 

In very simple cases of inverse reasoning, hypothesis 
may seem altogether needless. To take numbers again as 
a convenient illustration, I have only to look at the series, 

I, 2, 4, 8, 16, 32, &c., 
to know at once that the general law is that of geo- 
metrical progression ; I need no successive trial of various 
hypotheses, because I am familiar with the series, and have 
long since learnt from what general formula it proceeds. 
In the same way a mathematician becomes acquainted 
with the integrals of a number of common formulas, so 
that he need not go through any process of discovery. 
But it is none the less true that whenever previous reason- 
ing does not furnish the knowledge, hypotheses must be 
framed and tried (p. 124). 

There naturally arise two cases, according as the nature 
of the subject admits of certain or only probable deductive 
reasoning. Certainty, indeed, is but a singular case of 
probability, and the general principles of procedure are 
always the same. Nevertheless, when certainty of infer- 
ence is possible, the process is simplified. Of several 
mutually inconsistent hypotheses, the results of which 
can be certainly compared with fact, but one hypothesis 
can ultimately be entertained. Thus in the inverse logical 
problem, two logically distinct conditions could not yield 
the same series of possible combinations. Accordingly, 
in the case of two terms we had to choose one of six 
different kinds of propositions (p. 136), and in the case of 
three terms, our choice lay among 192 possible distinct 
hypotheses (p. 140). Natural laws, however, are often 
quantitative in character, and the possible hypotheses are 
then infinite in variety. 

When deduction is certain, comparison with fact is 
needed only to assure ourselves that we have rightly 
selected the hypothetical conditions. The law establishes 
itself, and no number of particular verifications can add 
to its probability. Having once deduced from the prin- 
ciples of algebra that the difference of the squares of two 
numbers is equal to the product of their sum and dif- 
ference, no number of particular trials of its truth will 
render it more certain. On the other hand, no finite 

number of particular verifications of a supposed law will 
render that law certain. In short, certainty belongs only 
to the deductive process, and to the teachings of direct 
intuition ; and as the conditions of nature are not given 
by intuition, we can only be certain that we have got a 
correct hypotliesis when, out of a limited number con- 
ceivably possible, we select that one which alone agrees 
with the facts to be explained. 

In geometry and kindred branches of mathematics, 
deductive reasoning is conspicuously certain, and it would 
often seem as if the consideration of a single diagram 
yields us certain knowledge of a general proposition. . 
But in reality all this certainty is of a purely hypothetical 
character. Doubtless if we could ascertain that a sup- 
posed circle was a true and perfect circle, we could be 
certain concerning a multitude of its geometrical pro- 
perties. But geometrical figures are physical objects, and 
the senses can never assure us as to their exact forms. 
The figures really treated in Euclid's Mements are 
imaginary, and we never can verify in practice the 
conclusions which we draw with certainty in inference; 
questions of degree and probability enter. 

Passing now to subjects in which deduction is only 
probable, it ceases to be possible to adopt one hypothesis 
to the exclusion of the others. We must entertain at the 
same time all conceivable hypotheses, and regard each 
with the degree of esteem proportionate to its probability. 
We go through the same steps as before. 

(1) We frame an hypothesis. 

(2) We deduce the probability of various series of pos- 
sible consequences. 

(3) We compare the consequences with the particular 
facts, and observe the probability that such facts would 
happen under the hypothesis. 

The above processes must be performed for every con- 
ceivable hypothesis, and then the absolute probability of 
each will be yielded by the principle of the inverse 
method (p. 242). As in the case of certainty we accept 
that hypothesis which certainly gives the required results, 
so now we accept as most probable that hypothesis which 
most probably gives the results; but we are obliged to 
entwrtain at the some time all other hypotheses with 










degrees of probability proportionate to the probabilities 
that they would give the same results. 

So far we have treated only of the process by which 
we pass from special facts to general laws, that inverse 
application of deduction which constitutes induction. 
But the direct employment of deduction is often com- 
bined with the inverse. No sooner have we established 
a general law, than the mind rapidly draws particular 
consequences from it In geometry we may almost seem 
to infer that because one equilateral triangle is equiangular, 
therefore another is so. In reality it is not because one is' 
that another is, but because all are. The geometrical con- 
ditions are perfectly general, and by what is sometimes 
called parity of reasoning whatever is true of one equilateral 
triangle, so far as it is equilateral, is true of all equilateral 

Similarly, in all other cases of inductive inference, 
where we seem to pass from some particular instances to 
a new instance, we go through the same process. We 
form an hypothesis as to the logical conditions under 
which the given instances might occur; we calculate 
inversely the probability of that hypothesis, and com- 
pounding this with the probability that a new instance 
would proceed from the same conditions, we gain the 
absolute probability of occurrence of the new instance in 
virtue of this hypothesis. But as several, or many, or 
even an. infinite number of mutually inconsistent hypo- 
theses may be possible, we must repeat the calculation for 
each such conceivable hypothesis, and then the complete 
probability of the future instance will be the sum of the 
separate probabilities. The complication of this process 
is often very much reduced in practice, owing to the fact 
that one hypothesis may be almost certainly true, and 
other hypotheses, though conceivable, may be so im- 
probable as to be neglected without appreciable error. 

When we possess no knowledge whatever of the con- 
ditions from which the events proceed, we may be unable 
to form any probable hypotheses as to their mode of 
origin. We have now to fall back upon the general 
solution of the problem effected by Laplace, which consists 
in admitting on an equal footing every conceivable ratio 
of favourable and unfavourable diances for the production 


of the event, and then accepting the aggregate result as 
the best which can be obtained This solution is only to 
be accepted in the absence of all better means, but like 
other results of the calculus of probability, it comes to our 
aid where knowledge is at an end and ignorance begins 
and It prevents us from over-estimating the knowledge we 
possess. The general results of the solution are in accord- 
ance with common sense, namely, that the more often an 
event has happened the more probable, as a general rule 
IS Its subsequent recurrence. With the extension of 
experience this probabiHty increases, but at the same time 
the probability is slight that events will long continue to 
happen as they have previously happened. 

We have now pursued the theory of inductive inference 
as far as can be done with regard to simple logical or 
numencal relations. The laws of nature deal with time 
and space, which are infinitely divisible. As we passed 
from pure logic to numerical logic, so we must now pass 
from questions of discontinuous, to questions of continuous 
quantity, encountering fresh considerations of much dif- 
ficulty. Before, therefore, we consider how the great in- 
ductions and generalisations of physical science illustrate 
the views of mductive reasoning just explained, we must 
break off for a time, and review the means which we 
possess of measuring and comparing magnitudes of time 
space mass, force, momentum, energy, and the various 
manifestations of energy in motion, heat, electricity, 
chemical change, and the other phenomena of nature 


r' I 






' ; 








As physical science advances, it becomes more and 
more accurately quantitative. Questions of simple logical . 
fact after a time resolve themselves into questions of 
degree, time, distance, or weight. Forces hardly suspected 
to exist by one generation, are clearly recognised by the 
next, and precisely measured by the third generation, , 
But one condition of this rapid advance is the invention 
of suitable instruments of measurement. We need what 
Francis Bacon called InstarUias citantes, or evocantes, 
methods of rendering minute phenomena perceptible to 
the senses ; and we also require Instantice radii or curri- 
evliy that is measuring instruments. Accordingly, the 
introduction of a new instrument often forms an epoch in 
the history of science. As Davy said, " Nothing tends so 
much to the advancement of knowledge as the application 
of a new instrument. The native intellectual powers of 
men in different times are not so much the causes of the 
different success of their labours, as the peculiar nature 
of the means and artificial resources in their possession." 
In the absence indeed of advanced theory and analyti 




cal power, a very precise instrument would be useless. 
Measuring apparatus and mathematical theory should didi- 
vmcQ pari passu, and with just such precision as the theorist 
can anticipate results, the experimentalist should be able 
to compare them with experience. The scrupulously 
accurate observations of Flamsteed were the proper 
complement to the intense mathematical powers of 

Every branch of knowledge commences with quantita- 
tive notions of a very rude character. After we have far 
progressed, it is often amusing to look back into the 
infancy of the science, and contrast present with past 
methods. At Greenwich Observatory in the present day, 
the hundredth part of a second is not thought an in- 
considerable portion of time. The ancient Chaldeans 
recorded an eclipse to the nearest hour, and the early 
Alexandrian astronomers thought it superfluous to dis- 
tmguish between the edge and centre of the sun. By 
the introduction of the astrolabe, Ptolemy and the latei 
Alexandrian astronomers could determine the places of 
the heavenly bodies within about ten minutes of arc 
Little progress then ensued for thirteen centuries, until 
Tycho Brahe made the first great step towards accuracy, 
not only by employing better instruments, but even 
more by ceasing to regard an instrument as correct 
Tycho, in fact, determined the errors of his instruments, 
and corrected his observations. He also took notice' 
of the effects of atmospheric refraction, and succeeded 
m attaining an accuracy often sixty times as great as 
that of Ptolemy. Yet Tycho and Hevelius often erred 
several minutes in the determination of a starts place, and 
It was a great achievement of Roemer and Flamsteed to 
reduce this error to seconds. Bradley, the modern Hip- 
parchus, carried on the improvement, his errors in right 
ascension, according to Bessel, being under one second of 
time, and those of declination under four seconds of arc. 
In the present day the average error of a single observa- 
tion is probably reduced to the half or quarter of what it 
was in Bradley's time; and further extreme accuracy is 
^t^^^^ ^y ^e multiplication of observations, and their 
skilful combination according to the theory of error. 
Some of the more important constants, for instance that 




i i> ' 





H^H .' 

rM ) 













of nutation, have been determined within the tenth part 
of a second of space.^ 

It would be a matter of great interest to trace out the 
dependence of this progress upon the introduction of 
new instruments. The astrolabe of Ptolemy, the tele- 
scope of Galileo, the pendulum of Galileo and Huyghens, 
the micrometer of Horrocks, and the telescopic sights and 
micrometer of Gascoygne and Picard, Kcemer's transit in- 
strument, Newton's and Hadley's quadrant, Dollond's 
achromatic lenses, Harrison's chronometer, and Ramsden's 
dividing engine — such were some of the principal addi- 
tions to astronomical apparatus. The result is, that we 
now take note of quantities, 300,000 or 400,000 times as 
small as in the time of the Chaldseaus. 

It would be interesting again to compare the scrupulous 
accuracy of a modem trigonometrical survey with Erato- 
sthenes' rude but ingenious guess at the difference of lati- 
tude between Alexandria and Syene — or with Norwood's 
measurement of a degree of latitude iu 1635. " Sometimes 
I measured, sometimes I paced," said Norwood ; " and I 
believe I am within a scantling of the truth." Such was 
the germ of those elaborate geodesical measurements 
which have made the dimensions of the globe known to 
us within a few hundred yards. 

In other branches of science, the invention of an instru- 
ment has usually marked, if it has not made, an epoch. 
The science of heat might be said to commence with the 
construction of the thermometer, and it has recently been 
advanced by the introduction of the thermo-electric pile. 
Chemistry has been created chiefly by the careful use of 
the balance, •which forms a unique instance of an instru- 
ment remaining substantially in the form in which it was 
first applied to scientific purposes by Archimedes. The 
balance never has been and probably never can be im- 
proved, except in details of construction. The torsion 
balance, introduced by Coulomb towards thf* end of last 
century, has rapidly become essential in many branches 
of investigation. In the hands of Cavendish and Baily, it 
gave a determination of the earth's density ; applied in the 
galvanometer, it gave a delicate measure of electrical 

* Baily, British Association Catalogue of Stars, pp. 7, 23. 




forces, and is indispensable in the thermo-electric pDa 
This balance is made by simply suspending any light rod 
by a thin \vire or thread attached to the middle point. 
And we owe to it almost all the more delicate investiga- 
tions in the theories of heat, electricity, and magnetism." 

Though we can now take note of the millionth of an 
inch in space, and the millionth of a second in time, we 
must not overiook the fact that in other operations of 
science we are yet in the position of the Chaldteans. Not 
many years have elapsed since the magnitudes of the 
stars, meaning the amounts of light they send to the 
observer's eye, were guessed at in the rudest manner, and 
the astronomer adjudged a star to this or that order of 
magnitude by a rough comparison with other stars of the 
same order. To Sir John Herschel we owe an attempt 
to introduce a uniform method of measurement and 
expression, bearing some relation to the real photometric 
magnitudes of the stars.^ Previous to the researches 
of Bunsen and Roscoe on the chemical action of light, 
we were devoid of any mode of measuring the energy of 
light ; even now the methods are tedious, and it is not 
clear that they give the energy of light so much as one of 
its special effects. Many natural phenomena have hardly 
yet been made the subject of measurement at all, such 
as the intensity of sound, the phenomena of taste and 
smell, the magnitude of atoms, the temperature of the 
electric spark or of the sun's photosphere. 

To suppose, then, that quantitative science ti-eats only of 
exactly measurable quantities, is a gross if it be a common 
mistake. Whenever we are treating of an event which 
either happens altogether or does not happen at all, we are 
engaged with a non-quantitative phenomenon, a matter of 
fact, not of degree ; but whenever a thing may be greater or 
less, or twice or thrice as great as another, whenever, in 
short, ratio enters even in the rudest manner, there 
science will have a quantitative character. There can 
be Uttle doubt, indeed, that every science as it pro- 
firesses will become gradually more and moio quantita- 
tive. Numerical precision is the soul of science, as 

* Outlines of Astronomy, 4th ed. sect. 781, p. 522 
Observations at the Cape of Good Hope, &c., p. 371 

Results of 








Herschel said, and as all natural objects exist in flpace, afid 
involve molecular movements, measurable in velocity and 
extent, there is no apparent limit to the ultimate extension 
of quantitative science. But the reader must not for a 
moment suppose that, because we depend more and more 
upon mathematical methods, we leave logical methods 
behind us. Number, as I have endeavoured to show, is 
logical in its origin, and quantity is but a development of 
number, or analogous thereto. 

Division of the Subject, 

The genei-al subject of quantitative investigation will 
have to be divided into several parts. We shall firstly 
consider the means at our disposal for measuring phe- 
nomena, and thus rendering them more or less amenable 
to mathematical treatment This task will involve an 
analysis of the principles on which accui-ate methods of 
measurement are founded, forming the subject of the 
remainder of the present chapter. As measurement, how- 
ever, only yields ratios, we have in the next chapter to 
consider the establishment of unit magnitudes, in terms of 
which our results may be expressed. As every pheno- 
menon is usually the sum of several distinct quantities 
depending upon different causes, we have next to investi- 
gate in Cliapter XV. the methods by which we may disen- 
tangle complicated effects, and refer each part of the joint 
effect to its separate cause. 

It yet remains for us in subsequent chapters to treat of 
quantitative induction, properly so called. We must 
follow out the inverse logical method, as it presents itself 
in problems of a far higher degree of difficulty than those 
which treat of objects related in a simple logical manner, 
and incapable of merging into each other by addition and 

Cowlinuous Quantity, 

The phenomena of nature are for the most part mani- 
fested in quantities which increase or decrease continu- 
ously. When we inquire into the precise meaning of 
continuous quantity, we find that it can only be described 




as that which is divisible without limit. We can divide 
a millimetre mto ten, or a hundred, or a thousand, or ten 
thousand parts, and mentally at any rate we can carry 
on the division ad infinUum. Any finite space, then 
must be conceived as made up of an infinite number of 
parts each infinitely small. We cannot entertain the 
simplest geometrical notions without allowing this The 
conception of a square involves the conception of *a side 
and diagonal, which, as Euclid beautifully proves in the 
117th proposition of his tenth book, have no common 
measure,! meaning no finite common measure. Incom- 
mensurable quantities are, in fact, those which have for their 
only common measure an infinitely small quantity It is 
somewhat startling to find, too, that in theory incommen- 
surable quantities will be infinitely more frequent than 
commensurable. Let any two lines be drawn haphazard ; 
It is infinitely unlikely that they will be commensurable 
so that the commensurable quantities, which we are sup- 
posed to deal with in practice, are but singular cases 
among an infinitely greater number of incommensurable 

Practically, however, we ti-eat all quantities as made up 
of the least quantities which our senses, assisted by the 
best measunng instruments, can perceive. So long as 
microscopes were uninvented, it was sufficient to regard 
an inch as made up of a thousand thousandths of an 
inch; now we must treat it as composed of a million 
mmionths. We might apparently avoid all mention of 
mlmitely small quantities, by never carrying our approxi- 
mations beyond quantities which the senses can appreciate 
In geometry, as thus treated, we should never assert two 
quantities to be equal, but only to be apparently equal. c\ 
Legendre really adopts this mode of treatment in the 
twentieth proposition of the first book of his Geometry • 
and It is practically adopted throughout the physical 
sciences, as we shall afterwards see. But though our 
nngers, and senses, and instruments must stop somewhere, 
there is no reason why the mind should not go on. W« 
can see that a proof which is only carried through a few 
Bteps in fact, might be carried on without limit, and it i« 

• Sec De Moi^gan, Study of Mathematics, in V.K.Q. Library, p. 8s 

T 2 


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'I i I 









this consciousness of no stopping-place, which renders 
Euclid's proof of his 117th proposition so impressive. Try 
how we will to circumvent the matter, we cannot really 
avoid the consideration of the infinitely small and the 
infinitely great. The same methods of approximation 
which seem confined to the finite, mentally extend them- 
selves to the infinite. 

One result of these considerations is, that we cannot 
possibly adjust two quantities in absolute equality. The 
suspension of Mahomet's coffin between two precisely 
equal magnets is theoretically conceivable but practically 
impossible. The story of the Merchant of Venice turns 
upon the infinite improbability that an exact quantity of 
liesh could be cut. Unstable equilibrium cannot exist in 
nature, for it is that which is destroyed by an infinitely 
small displacement. It might be possible to balance an 
egg on its end practically, because no egg has a surface of 
perfect curvature. Suppose the egg shell to be perfectly 
smooth, and the feat would become impossible. 

T/ie Fallacious Indications of tlu Senses. 

I may briefly remind the reader how little we can trust 
to our unassisted senses in estimating the degree or 
magnitude of any phenomenon. The eye cannot correctly 
estimate the comparative brightness of two luminous 
bodies which differ much in brilliancy ; for we know 
that the iris is constantly adjusting itself to the intensity 
of the light received, and thus admits more or less light 
according to circumstances. Tlie moon which shines vnth 
almost dazzling brightness by night, is pale and nearly 
imperceptible while the eye is yet affected by the vastly 
more powerful light of day. Much has been recorded 
concerning the compamtive brightness of the zodiacal 
light at different times, but it would be difficult to prove 
that these changes are not due to the varying darkness 
at the time, or the different acuteness of the observer's 
eye. For a like reason it is exceedingly difficult to esta- 
blish the existence of any change in the form or compara- 
tive brightness of nebulae; the appearance of a nebula 
greatly depends upon the keenness of sight of the 
observer, or the accidental condition of freshness or 




fatigue of his eya The same is true of lunar obser- 
vations; and even the use of the best telescope fails 
to remove this difficulty. In judging of colours, again, 
we must remember that light of any given colour tends 
to dull the sensibility of the eye for light of the same 

Nor is the eye when unassisted by instruments a much 
better judge of magnitude. Our estimates of the size of 
minute bright points, such as the fixed stars, are com- 
pletely falsified by the effects of irradiation. Tycho 
calculated from the apparent size of the star-discs, that 
no one of the principal fixed stars could be contained 
within the area of the earth's orbit. Apart, however, from 
irradiation or other distinct causes of error our visual 
estimates of sizes and shapes are often astonishingly 
incorrect Artists almost invariably dmw distant moun- 
tains in ludicrous disproportion to nearer objects, as a 
comparison of a sketch with a photograph at once shows. 
The extraordinary apparent difference of size of the sun 
or moon, according as it is high in the heavens or near 
the horizon, should be sufficient to make us cautious in 
accepting the plainest indications of our senses, unassisted 
by instrumental measurement As to statements concern- 
ing the height of the aurora and the distance of meteors, 
they are to be utterly distrusted. When Captain Parry 
says that a ray of the aurora shot suddenly downwards 
between him and the (land which was only 3,000 yards 
distant, we must consider him subject to an illusion of 
sense. ^ 

It is true that errors of observation are more often 
errors of judgment than of sense. That which is actually 
seen must be so far truly seen ; and if we correctly interpret 
the meaning of the phenomenon, there would be no error 
at all But the weakness of the bare senses as measuring 
instruments, arises from the fact that they import varying 
conditions of unknown amount, and we cannot make the 
requisite corrections and allowances as in the case of a 
solid and invariable instrument 

Bacon has excellently stated the insufficiency of the 

' I««>mi8, On th€ Aurora Borealis, Smithsonian Transactiona. 
quoting Parry's Third Voyage, p. 61. 




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I ■ 


J I 




senses for estimating the magnitudes of objects, or de- 
tecting the degrees in which phenomena present them- 
selves. " Tilings escape the senses/' he says, " because the 
object is not sufficient in quantity to strike the sense : as 
all minute bodies ; because the percussion of the object is 
too great to be endured by the senses: as the form of the 
sun when looking directly at it in mid-day ; because the 
time is not proportionate to actuate the sense: as the 
motion of a bullet in the air, or the quick circular motion 
of a firebrand, which are too fast, or the hour-hand of 
a common clock, which is too slow ; from the distance 
of the object as to place: as the size of the celestial 
bodies, and the size and nature of all distant bodies; 
from prepossession by another object : as one powerful 
smell renders other smells in the same room imper- 
ceptible ; from the interruption of interposing bodies : 
as the internal parts of animals ; and because the object 
is unfit to make an impression upon the sense : as the 
air or the invisible and untangible spirit which is in- 
cluded in every living body." 

Complexity of Quantitative Questions. 

One remark which we may well make in entering 
upon quantitative questions, has regard to the great variety 
and extent of phenomena presented to our notica So 
long as we deal only with a simply logical question, that 
question is merely, Does a certain event happen ? or. Does 
a certain object exist ? No sooner do we regard the event 
or object as capable of more and less, than the question 
branches out into many. We must now ask, How much 
18 It compared with ite cause ? Does it change when the 
amount of the cause changes ? If so, does it change in 
the same or opposite direction ? Is the change in simple 
proportion to that of the cause ? If not, what more com- 
plex law of connection holds true ? This law determined 
satisfactorily in one series of circumstances may be varied 
under new conditions, and the most complex relations of 
several quantities may ultimately be established. 

In every question of physical science there is thus a 
series of steps the first one or two of which are usually 
TPftde with ease while the succeeding ones demand more 




and more careful measurement. We cannot lay down 
any invariable series of questions which must be asked 
from nature. The exact character of the questions will 
vary according to the nature of the case, but they will 
usually be of an evident kind, and we may readily illus- 
trate them by examples. Suppose that we are investigat- 
ing the solution of some salt in water. The first is a 
purely logical question : Is there solution, or is there not ? 
Assuming the answer to be in the affirmative, we next 
inquire, Does the solubility vary with the temperature, or 
not ? In all probability some variation will exist, and we 
must have an answer to the further question. Does 
the quantity dissolved increase, or does it diminish with 
the temperature? In by far the greatest number of 
cases salts and substances of all kinds dissolve more freely, 
the higher the temperature of the water ; but there are a 
few salts, such as calcium sulphate, which follow the 
opposite rule. A considerable number of salts resemble 
sodium sulphate in becoming more soluble up to a certain 
temperature, and then varying in the opposite direction. 
We next require to assign the amount of variation as 
compared with that of the temperature, assuming at first 
that the increase of solubility is proportional to the in- 
crease of temperature. Common salt is an instance of 
very slight variation, and potassium nitrate of very con- 
siderable increase with temperature. Accurate observa- 
tions will probably show, however, that the simple law 
of proportionate variation is only approximately true, 
and some more complicated law involving the second, 
third, or higher powers of the temperature may ultimately 
be established. All these investigations have to be 
carried out for each salt separately, since no distinct prin- 
ciples by which we may infer from one substance to 
another have yet been detected. There is still an in- 
definite field for further research open ; for the solubility 
k-^^*^ will probably vary with the pressure under 
which the medium is placed ; the presence of other salts 
already dissolved may, have effects vet unknown. The 
researches already elfected as regards the solvent power of 
water must be repeated with alcohol, ether, carbon 
bisulphide, and other media, so that unless general laws 
can be detected, this one phenomenon of solution can 





it. ' 



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never be exhaustively treated. The same kind of 'questions 
recur as regards the solution or absorption of gases in 
liquids, the pressure as well as the temperature having 
then a most decided effect, and Professor Roscoe's re- 
searches on the subject present an excellent example of 
the successive determination of various complicated laws.* 

There is hardly a branch of physical science in which 
similar complications are not ultimately encountered. 
In the case of gravity, indeed, we arrive at the final 
law, that the force is the same for all kinds of matter, 
and varies only with the distance of action. But in 
other subjects the laws, if simple in their ultimate nature, 
are disguised and complicated in their apparent results. 
Thus the effect of heat in expanding solids, and the reverse 
effect of forcible extension or compression upon the tem- 
perature of a body, will vary from one substance to 
auother, will vary as the temperature is already higher or 
lower, and«.will probably follow a highly complex law, 
which in some cases gives negative or exceptional results. 
In crystalline substances the same researches have to be 
repeated in each distinct axial direction. 

In the sciences of pure observation, such as those of 
astronomy, meteorology, and terrestrial magnetism, wo 
meet with many interesting series of quantitative deter- 
minations. The so-called fixed stars, as Giordano Bruno 
divined, are not really fixed, and may be more truly 
described as vast wandering orbs, each pursuing its own 
path through space. We must then determine separately 
for each star the following questions : — 

1. Does it move ? 

2. In what direction ? 

3. At what velocity ? 

4. Is this velocity variable or uniform ? 

5. If variable, according to what law ? 

6. Is the direction imiform ? 

7. If not, what is the form of the apparent path ? 

8. Does it approach or recede ? 

9. What is the form of the real path ? 

The successive answers to such questions in the case of 
certain binary stars, have aflforded a proof that the 

* WatU' IHctwuary of Chemistry, voL ii. p. 79a 




motions are due to a central force coinciding in law with 
gravity, and doubtless identical with it. In other cases 
the motions are usually so small that it is exceedingly 
difficult to distinguish them with certainty. And the time 
is yet far off when any general results as regards stellar 
motions can be established. 

The variation in the brightness of stars opens an un- 
limited field for curious observation. There is not a star 
in the heavens concerning which we might not have to 
determine : — 

I. Does it vary in brightness ? 

a. Is the brightness increasing or decreasing ? 

3. Is the variation uniform ? 

4. If not, acording to what law does it vary ? 

In a majority of cases the change will probably be 
found to have a periodic character, in which case several 
other questions will arise, such as — 

5. What is the length of the period ? 

6. Are there minor periods ? 

7. What is the law of variation within the period ? 

8. Is there any change in the amount of variation ? 

9. If so, is it a secular, i.e. a continually growing 
change, or does it give evidence of a greater period ? 

Already the periodic changes of a certain number of 
stars have been determined with accuracy, and the lengths 
of the periods vary from less than three days up to 
intervals of time at least 250 times as great. Periods 
within periods have also been detected. 

There is, perhaps, no subject in which more complicated 
quantitative conditions have to be determined than ter- 
restrial magnetism. Since the time when the declination 
of the compass was first noticed, as some suppose by 
Columbus, we have had successive discoveries from time 
to time of the progressive change of declination from 
century to century; of the periodic character of this 
change; of the difference of th§ declination in various 
parts of the earth's surface; of the varying laws of 
the change of declination ; of the dip or inclination of 
the needle, and the corresponding laws of its periodic 
changes ; the horizontal and perpendicular intensities have 
also been the subject of exact measurement, and have been 
found to vary with place and time, like the directions of 

' 1 


•I ; 






the needle ; daily and yearly periodic changes have also 
been detected, and all the elements are found to be subject 
to occasional storms or abnormal perturbations, in which 
the eleven year period, now known to be common to many 
planetary relations, is apparent The complete solution 
of these motions of the compass needle involves nothing 
less than a determination of its position and oscillations in 
every part of the world at any epoch, the like determina- 
tion for another epoch, and so on, time after time, until 
the periods of all changes are ascertained. This one sub- 
ject offers to men of science an almost inexhaustihle field 
for interesting quantitative research, in which we shall 
doubtless at some future time discover the operation of 
causes now most mysterious and unaccountable. 

The Methods of Acmrate Measurement. 


In studying the modes hy which physicisU have ac- 
complished very exact measurements, we find that they 
are very various, but that they may perhaps be reduced 
under the following three classes : — 

I. The increase or decrease, in some determinate ratio, 
of the quantity to be measured, so as to bring it within 
the scope of our senses, and to equate it with the standard 
unit, or some determinate multiple or sub-multiple of this 

a. The discovery of some natural conjunction of events 
which will enable us to compare directly the multiples of 
the quantity with those of the unit, or a quantity related 
in a definite ratio to that unit 

3. Indirect measurement, which gives us not the quan- 
tity itself, but some other quantity connected with it by 
known mathematical relations. 

Conditions of Acmrate Measurement. 

Several conditions are requisite in order that a mea- 
surement may be made with great accuracy, and that 
the results may be closely accordant when several inde- 
pendent measurements are made. 

In the first place the magnitude must be exactly defined 
'^ by sharp terminations, or precise marks of inconsiderable 




thickness. When a boundary is vague and graduated, 
like the penumbra in a lunar eclipse, it is impossible to 
say where the end really is, and different people will come 
to different results. We may sometimes overcome this 
difficulty to a certain extent, by observations repeated in 
a special manner, as we shall afterwards see ; but when 
possible, we should choose opportunities for measure- 
ment when precise definition is easy. The moment of 
occultation of a star by the moon can be observed with 
great accuracy, because the star disappears with perfect 
suddenness ; but there are other astronomical conjunctions, 
eclipses, transits, &c., which occupy a certain length of 
time in happening, and thus open the way to differences 
of opinion. It would be impossible to observe with pre- 
cision the movements of a body possessing no definite 
points of reference. The colours of the complete spectrum 
shade into each other so continuously that exact deter- 
minations of refractive indices would have been impossible, 
had we not the dark lines of the solar spectrum as precise 
points for measurement, or various kinds of homogeneous 
light, such as that of sodium, possessing a nearly uniform 
length of vibration. 
/p In the second place, we cannot measure accurately 
unless we have the means of multiplying or dividing 
a quantity without considerable error, so that we may 
correctly equate one magnitude with the multiple or sub- 
multiple of the other. In some cases we operate upon the 
quantity to be measured, and bring it into accurate coin- 
cidence with the actual standard, as when in photometry 
we vary the distance of our luminous body, until its 
illuminating power at a certain point is equal to that of a 
standard lamp. In other cases we repeat the unit until it 
equals the object, as in surveying land, or determining a 
weight by the balance. The requisites of accuracy now 
are :— (i) That we can repeat unit after unit of exactly 
equal magnitude ; (2) That these can be joined together 
so that the aggregate shall really be the sum of the 
parts. The same conditions apply to subdivision, which 
may be regarded as a multiplication of subordinate units. 
In order to measure to the thousandth of an inch, we must 
be able to add thousandth after thousandth without error 
in the magnitude of these spaces, or in their conjunction. 



ft ', il 






Measuring InstrumenU, 

To consider the mechanical construction of scientific 
instruments, is no part of my purpose in this book. I 
wish to point out merely the general purpose of sucli 
instruments, and the methods adopted to cany out that 
purpose with great precision. In the first place we must 
distinguish between the instrument which effects a com- 
parison between two quantities, and the standard mag- 
nitude which often forms one of the quantities compared. 
The astronomer's clock, for instance, is no standard of the 
efflux of time; it serves but to subdivide, with approxi- 
mate accuracy, the interval of successive passages of a 
star across the meridian, which it may effect perhaps to 
the tenth part of a second, or agAoo part of the whole. 
The moving globe itself is the real standard clock, and the 
transit instrument the finger of the clock, while the stars 
are the hour, minute, and second marks, none the less 
accurate because they are disposed at unequal intervals. 
The photometer is a simple instrument, by which we com- 
pare the relative intensity of rays of light falling upon a 
given spot. The galvanometer shows the comparative 
intensity of electric currents passing through a wire. 
The calorimeter gauges the quantity of heat passing from 
a given object But no such instruments furnish the 
standard unit in terms of which our results are to be ex- 
pressed. In one peculiar case alone does the same instru- 
ment combine the unit of measurement and the means of 
comparison. A theodolite, mural circle, sextant, or other 
instrument^for the measurement of angular magnitudes 
has no need of an additional physical unit ; for the circle 
itself, or complete revolution, is the natural unit to which 
all greater or lesser amounts of angular magnitude are 

The result of every measurement is to make known the 
purely numerical ratio existing between the magnitude 
to be measured, and a certain other magnitude, which 
should, when possible, be a fixed unit or standard magni- 
tude, or at least an intennediate unit of which the value 
can be ascertained in terms of the ultimate standard. But 
though a ratio is the required result, an equation is the 
mode in which the ratio is determined and expressed. In 




every measurement we equate some multiple or submul- 
tiple of one quantity, with some multiple or submultiple 
of another, and equality is always the fact which we 
ascertain by the senses. By the eye, the ear, or the touch, 
we judge whether there is a discrepancy or not between 
two lights, two sounds, two intervals of time, two bars of 
metal Often indeed we substitute one sense for the other, 
as when the efflux of time is judged by the marks upon 
a moving slip of paper, so that equal intervals of time are 
represented by equal lengths. There is a tendency to 
reduce all comparisons to the comparison of space magni- 
tudes, but in every case one of the senses must be the 
ultimate judge of coincidence or nou-coincidence. 

Since the equation to be established may exist between 
any multiples or submultiples of the quantities compared, 
there natumlly arise several different modes of comparison 
adapted to different cases. Let p be the magnitude to 
be measured, and q that in terms of which it is to be 
expressed. Then we wish to find such numbers x and y, 

that the equation p — - q may be tyue. This equation 

may be presented in four forms, namely :— 

First Form. 

Second Form. 

Third Form. 

Fourth Form. 

,= ., 

!>; = « 


? = i 

Each of these modes of expressing the same equation cor- 
responds to one mode of effecting a measurement 

When the standard quantity is greater than that to be 
measured, we often adopt the first mode, and subdivide 
the unit until we get a magnitude equal to that measured. 
The angles observed in surveying, in astronomy, or in 
goniometry are usually smaller than a whole revolution, 
and the measuring circle is divided by the use of the 
screw and microscope, until we obtain an angle undistin- 
guishable from that observed. The dimensions of minute 
objects are determined by subdividing the inch or centi- 
metre, the screw micrometer being the most accurate 
means of subdivision. Ordinary temperatures are esti- 
mated by division of the standard interval between the 
freezing and boiling points of water, as marked on a 
thermometer tube. 


•; I 





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In a still greater number of cases, perhaps, we multiply 
the standard unit until we get a magnitude equal to that 
to be measured. Ordinary measurement by a foot rule, 
a surveyor's chain, or the excessively careful measurements 
of the base line of a trigonometiical survey by standard 
bars, are sufficient instances of this procedure. 

In the second case, where p - = 5, we multiply or divide 

a magnitude until we get what is equal to the unit, or to 
some magnitude easily comparable with it As a general 
rule the quantities which we desire to measure in 
physical science are too small rather than too great for 
easy determination, and the problem consists in multiply- 
ing them without introducing error. Thus the expansion 
of a metallic bar when heated from o* C to 100° may be 
multiplied by a train of levers or cog wheels. In the 
common thermometer the expansion of the mercury, 
though slight, is rendered very apparent, and easily 
measurable by the fineness of the tube, and many other 
cases might be quoted. There are some phenomena, on 
the contrary, which are too great or rapid to come within 
the easy range of our senses, and our task is then the oppo- 
site one of diminution. Galileo found it difficult to measure 
the velocity of a falling body, owing to the considerable 
velocity acquired in a single second. He adopted the 
el^ant device, therefore, of lessening the rapidity by 
letting the body roll down an inclined plane, which 
enables us to reduce the accelerating force in any required 
ratio. The same purpose is effected in the well-known 
experiments performed on Attwood's machine, and the 
measurement of gravity by the pendulum really depends 
on the same principle applied in a far more advantageous 
manner. Wheatstone invented a beautiful method of gal- 
vanometry for strong currents, which consists in drawing 
off from the main current a certain determinate portion, 
which is equated by the galvanometer to a standard 
current In short, he measures not the current itself but 
a known fraction of it 

In many electrical and other experiments, we wish to 
measure the movements of a needle or other body, which 
are not only very slight in themselves, but the manifes- 
tations of exceedingly small forces. We camiot even 




approach a delicately balanced needle without disturbing 
it Under these circumstances the only mode of proceed- 
ing with accuracy, is to attach a very small mirror to the 
moving body, and employ a ray of light reflected from 
the mirror as an index of its movemenfeg. The ray may 
be considered quite incapable of affecting the body, and 
yet by allowing the ray to pass to a sufficient distance, 
the motions of the mirror may be increased to almost any 
extent A ray of light is in fact a perfectly weightless 
finger or index of indefinite length, with the additional 
advantage that the angular deviation is by the law of 
reflection double that of the mirror. This method was 
introduced by Gauss, and is now of great importance ; 
but in Wollaston's reflecting goniometer a ray of light 
had previously been employed as an index. "Lavoisier 
and Laplace had also used a telescope in connection with 
the pyrometer. 

It is a great advantage in some instruments that they 
can be readily made to manifest a phenomenon in a greater 
or less degree, by a very slight change in the construction. 
Thus either by enlarging the bulb or contracting the tube 
of the thermometer, we can make it give more conspicuous 
indications of change of temperature. The ordinary baro- 
meter, on the other hand, always gives the variations of 
pressure on one scale. The torsion balance is remark- 
able for the extreme delicacy which may be attained 
by mcreasing the length and lightness of the rod, and the 
length and thinness of the supporting thread. Forces so 
minute as the attraction of gravitation between two balls, 
or the magnetic and diamagnetic attraction of common 
Aquids and gases, may thus be made apparent, and even 
measui-ed. The common chemical balance, too, is capable 
theoretically of unlimited sensibility. 

The third mode of measui-ement, which may be called 
the Method of Repetition, is of such great importance and 
mterest that we must consider it in a separate section. It 
consists in multiplying both magnitudes to be compared 
until some multiple' of the first is found to coincide very 
nearly with some multiple of the second. If the multipli- 
cation can be effected to an unlimited extent, without the 
mtroduction of countervailing errors, the accuracy with 
which the required ratio can be determined is unlimited 


J ' 

• I 



and we thus account for the extraordinary precision with 
which intervals of time in astronomy are compared to- 

The fourth mode of measurement, in which we equate 
submultiples of two magnitudes, is comparatively seldom 
employed, because it does not conduce to accuracy. In 
the photometer, perhaps, we may be said to use it ; we 
compare the intensity of two sources of light, by placing 
them both at such distances from a given surface, that the 
light falling on the surface is tolerable to the eye, and 
equally intense from each source. Since the intensity of 
light varies inversely as the square of the distance, the 
relative intensities of the luminous bodies are propor- 
tional to the squares of their distances. The equal in- 
tensity of two rays of similarly coloured light may be 
most accurately ascertained in the mode suggested by 
Arago, namely, by causing the rays to pass in opposite 
directions through two nearly flat lenses pressed together. 
There is an exact equation between the intensities of the 
beams when Newton's rings disappear, the ring created 
by one ray being exactly the complement of that created 
by the other. 

The Method of Repetition. 

The ratio of two quantities .can be determined with 
unlimited accuracy, if we can multiply both the object 
of measurement and the standard unit without error, and 
then observe what multiple of the one coincides or nearly 
coincides with some multiple of the other. Although per- 
fect coincidence can never be really attained, the error 
thus arising may be indefinitely reduced. For if the 
equation pi/ = qx be uncertain to the amount e, so 

that py = qx ± e^ then we have p = q -:k', and 

as we are supposed to be able to make y as great as we 
like without increasing the error e, it follows that we 
can make e -:- y sls small as we like, and thus approxi- 
mate within an inconsiderable quantity to the required 
ratio X -r y. 

This method of repetition is naturally employed when- 
ever quantities can be repeated, or repeat themselve* 




without error of juxtaposition, which is especially tho 
case with the motions of the earth and heavenly bodies. 
In determining the length of the sidereal day, we deter- 
mine the ratio between the earth's revolution round the 
sun, and its rotation on its own axis. We might ascertain 
the ratio by observing the successive passages of a star 
across the zenith, and comparing the interval by a good 
clock with that between two passages of the sun, the 
difference being due to the angular movement of the 
eaith round the sun. In such observations we should 
have an error of a considerable part of a second at each 
observation, in addition to the iiTegularities of the clock. 
But the revolutions of the earth repeat themselves day 
after day, and year after year, without the slightest in- 
terval between the end of one period and the beginning 
of another. The operation of multiplication is perfectly 
performed for us by nature. If, then, we can find an obser- 
vation of the passage of a star across the meridian a hun- 
dred years ago, that is of the interval of time between 
the passage of the sun and the star, the instrumental 
en-ors in measuiing this interval by a clock and telescope 
niay be greater than in the present day, but will be 
divided by about 36,524 days, and rendered excessively 
small It is thus that astronomei-s have been able to 
ascertain the ratio of the mean solar to the sidereal day 
to the 8th place of decimals (100273791 to i), or to the 
hundred millionth pait, probably the most accurate result 
of measui-ement in the whole range of science. 

The antiquity of this mode of comparison is almost as 
great as that of astronomy itself. Hipparchus made the 
first clear application of it, when he compared his own 
observations with those of Aristarchus, made 145 years 
previously, and thus ascertained the length of the year. 
This calculation may in fact be regarded as the earliest 
attempt at an exact determination of the constants of 
nature. The method is the main resource of astrono- 
mers; Tycho, for instance, detected the slow diminution 
of the obliquity of 'the earth's axis, by the comparison 
of observations at long intervals. Living astronomers 
use the method as much as earlier ones; but so superior 
m accuracy are all observations taken during the last 
hundred years to aU previous ones, that it is often 










I I 

( ' 




found preferable to take a shorter interval, rather than 
incur the risk of greater instrumental errors in the earlier 

It is obvious that many of the slower changes of the 
heavenly bodies must require the lapse of large intervals 
of time to render their amount perceptible. Hipparchus 
could not possibly have discovered the smaller inequalities 
of the heavenly motions, because there were no previous 
observations of sufficient age or exactness to exhibit them. 
And just as the observations of Hipparchus formed the 
starting-point for subsequent comparisons, so a large part 
of the labour of present astronomers is directed to record- 
ing the present state of the heavens so exactly, that future 
generations of astronomers may detect changes, which 
cannot possibly become known in the present age. 

The principle of repetition was very ingeniously em- 
ployed in an instrument first proposed by Mayer in 1767, 
and carried into practice in the Repeating Circle of Borda. 
The exact measurement of angles is indispensable, not 
only in astronomy but also in trigonometrical surveys, and 
the highest skill in the mechanical execution of the gradu- 
ated circle and telescope will not prevent terminal errors 
of considerable amount If instead of one telescope, the 
circle be provided with two similar telescopes, these may 
be alternately directed to two distant points, say the 
marks in a trigonometrical survey, so that the circle shall 
be turned through any multiple of the angle subtended 
by those marks, before the amount of the angular revolu- 
tion is read off upon the graduated circle. Theoretically 
speaking, all error arising from imperfect graduation might 
thus be indefinitely reduced, being divided by the number 
of repetitions. In practice, the advantage of the invention 
is not found to be very great, probably because a certain 
error is introduced at each observation in the changing 
and fixing of the telescopes. It is moreover inapplicable 
to moving objects like the heavenly bodies, so that its use 
is confined to important trigonometrical surveys. 

The pendulum is the most perfect of all instruments, 
chiefly because it admits of almost endless repetition. 
Since the force of gravity never ceases, one swing of the 
pendulum is no sooner ended than the other is begun, 
80 that the juxtaposition of successive units is absolutely 




perfect. Provided that the oscillations be equal, one 
thousand oscillations will occupy exactly one thousand 
times as great an interval of time as one oscillation. 
Not only is the subdivision of time entirely dependent 
on this fact, but in the accurate measurement of gravity, 
and many other important determinations, it is of the 
greatest service. In the deepest mine, we could not 
observe the rapidity of fall of a body for more than a 
quarter of a minute, and the measurement of its velocity 
would be difficult, and subject to uncertain errors from 
resistance of air, &c. In the pendulum, we have a body 
which can be kept rising and falling for many hours, in 
a medium entirely under our command or if desirable in 
a vacuum. Moreover, the comparative force of gravity at 
different points, at the top and bottom of a mine for 
instance, can be determined with wonderful precision, by 
comparing the oscillations of two exactly similar pendu- 
lums, with the aid of electric clock signals. 

To ascertain the comparative times of vibration of two 
pendulums, it is only requisite to swing them one in 
front of the other, to record by a clock the moment when 
they coincide in swing, so that one hides the other, and 
then count the number of vibrations until they again come 
to coincidence. If one pendulum makes m vibrations and 
the other w, we at once have our equation pn ^ qm ; 
which gives the length of vibration of either pendulum in 
terms of the other. This method of coincidence, embody- 
ing the principle of repetition in perfection, was employed 
with wonderful skill by Sir George Airy, in his experi- 
ments on the Density of the Earth at the Harton Colliery, 
the pendulums above and below being compared with 
clocks, which again were compared with each other by 
electric signals. So exceedingly accurate was this method 
of observation, as earned out by Sir George Airy, that he 
was able to measure a total difference in the vibrations at 
the top and bottom of the shaft, amounting to only 2-24 
seconds in the twenty-four hours, with an error of less 
than one hundredth part of a second, or one part in 
<>,o4o,ooo of the whole day.^ 
The principle of repetition has been elegantly applied 

» Philosophical Transa4iii(mt, (I856) vol 146, Part L p. 297. 

U 2 




^"i h 











in observing the motion of waves in water. 11 the canal 
in which the experiments are made be short, say twenty 
feet long, the waves will pass through it so rapidly that 
an observation of one length, as practised by Walker, will 
be subject to much terminal error, even when the observer 
is very skilful. But it is a result of the undulatory theory 
that a wave is unaltered, and loses no time by com- 
plete reflection, so that it may be allowed to tmvel back- 
wards and forwards in the same canal, and its motion, say 
through sixty lengths, or 1200 feet, may be observed with 
the same accuracy as in a canal 1200 feet long, with tlie 
advantage of greater uniformity in the condition of the 
canal and water.^ It is always desirable, if possible, to 
bring an experiment into a small compass, so that it 
may be well under command, and yet we may often 
by repetition enjoy at the same time the advantage of 
extensive trial. 

One reason of the great accuracy of weighing with a 
good balance is the fact, that weights placed in the same 
scale are naturally added together without the slightest 
error. There is no difficulty in the precise juxtaposition 
of two grams, but the juxtaposition of two metre mea- 
sures can only be effected with tolerable accuracy, by tlie 
use of microscopes and many precautions. Hence, the 
extreme trouble and cost attaching to the exact measure- 
ment of a base line for a survey, the risk of error entering 
at every juxtaposition of the measuring bars, and inde- 
fatigable attention to all the requisite precautions being 
necessary throughout the operation. 

Measurements hy Natural Coincidence, 

In certain cases a peculiar conjunction of circumstances 
enables us to dispense more or less with instrumental 
aids, and to obtain very exact numerical results in the 
simplest manner. The mere fact, for instance, that no 
human being has ever seen a different face of the moon 
from that familiar to us, conclusively proves that the 
period of rotation of the moon on its own axis is equal 

* Airy, On Tides and Wavetj Enoyclopaedia Mctropolitana, p. 345. 
Scott Russell, Britiih AtiocxatioH Report^ 1837, p. 432. 

to that of its revolution round the earth. Not only have 
we the repetition of these movements during 1000 or 
2000 years at least, but we have observations made for 
us at very remote periods, free from instrumental error, 
no instrument being needed. We learn that the seventh 
satellite of Saturn is subject to a similar law, because its 
light undergoes a variation in each revolution, owing to 
the existence of some dark tract of land ; now this failure 
of light always occurs while it is in the same position 
relative to Saturn, clearly proving the equality of the 
axial and revolutional periods, as Huygens perceived.^ 
A like peculiarity in the motions of Jupiter's fourth satel- 
lite was similarly detected by Maraldi in 17 13. 

Remarkable conjunctions of the planets may sometimes 
allow us to compare their periods of revolution, through 
great intervals of time, with much accuracy. Laplace in 
explaining the long inequality in the motions of Jupiter 
and Saturn, was assisted by a conjunction of these 
planets, obserA^ed at Cairo, towards the close of the 
eleventh century. Laplace calculated that such a con- 
junction must have happened on the 31st of October, a.d. 
1087 ; and the discordance between the distances of the 
planets as recorded, and as assigned by theory, was less 
than one-fifth part of the apparent diameter of the sun. 
This difference being less than the probable error of the 
early record, the theory was confirmed as far as facts 
were available.* 

Ancient astronomers often showed the highest inge- 
nuity in turning any opportunities of measurement which 
occurred to good account. Eratosthenes, as early as 
250 B.C., happening to hear that the sun at Syene, in 
Upper Egypt, was visible at the summer solstice at 'the 
bottom of a well, proving that it was in the zenith, pro- 
posed to determine the dimensions of the earth, by mea- 
suring the length of the shadow of a rod at Alexandria on 
the same day of the year. He thus learnt in a rude 
manner the differeiice of latitude between Alexandria and 
Syene and finding it to be about one fiftieth part of the 
whole circumference, he ascertained the dimensions of the 

luL^^lL ^^*^^''^^*^' PP- "7,n8. Laplace's. %*<^m«, tmus- 
• Grant's History of Phytieal Astronomy, p. 129. 






i ■ 







.' , 


earth within about one sixth part of the truth. The use 
of wells in astronomical observation appears to have been 
occasionally practised in comparatively recent times as 
by Flamsteed in 1679.' The Alexandrian astronomers 
employed the moon as an instrument of measurement 
in several sagacious modes. Wlien the moon is exactly 
half full, the moon, sun, and eai-th, are at the angles of a 
right-angled triangle. Aristarchus measured at such a 
time the moon's elongation from the sun, which gave him 
the two other angles of the triangle, and enabled him to 
judge of the comparative distances of tlie moon and sun 
from the earth. His result, though very rude, was far 
more accurate than any notions previously entertained, 
and enabled him to form some estimate of the comparative 
magnitudes of the bodies. Eclipses of the moon were 
very useful to Hipparchus in ascertaining the longtitude 
of the stars, which are invisible when the sun is above 
the horizon. For the moon when eclipsed must be 180° 
distant from the sun ; hence it is only requisite to measure 
the distance of a fixed star in longitude from the eclipsed 
moon to obtain with ease its angular distance from the 

lu later times the eclipses of Jupiter have served to 
measure an angle; for at the middle moment of the 
eclipse the satellite must be in the same straight line with 
the planet and sun, so that we can learn from the known 
laws of movement of the satellite the longitude of Jupiter 
as seen from the sun. If at the same time we measure 
the elongation or apparent angular distance of Jupiter 
from the sun, as seen from the earth, we have all the 
angles of the triangle between Jupiter, the sun, and the 
earth, and can calculate the comparative magnitudes of 
the sides of the triangle by trigonometry. 

The transits of Venus over the sun's face are other 
natural events which give most accurate measurements 
of the sun's parallax, or apparent difference of position 
as seen from distant points of the earth's surface. The 
sun forms a kind of background on which the place of 
the planet is marked, and serves as a measuring instru- 
ment free from all the errors of construction which affect 

' Baily*8 Account of Flamstud, p. lix. 

human instruments. The rotation of the eai-th, too, by 
variously affecting the apparent velocity of ingress or 
egress of Venus, as seen from different places, discloses 
the amount of the parallax. It has been suflSciently 
shown that by rightly choosing the moments of obser- 
vation, the planetary bodies may often be made to reveal 
their relative distance, to measure their own position, to 
record their own movements with a high degree of 
accuracy. With the improvement of astronomical instru- 
ments, such conjunctions become less necessary to the 
progress of the science, but it will always remain advan- 
tageous to choose those moments for observation when 
instrumental errors enter with the least effect. 

In other sciences, exact quantitative laws can occasion- 
ally be obtained without instrumental measurement, as 
when we learn the exactly equal velocity of sounds of 
different pitch, by observing that a peal of bells or a 
musical performance is heard harmoniously at any dis- 
tance to which the sound penetrates; this could not be 
the case, as Newton remarked, if one sound overtook 
the other. One of the most important principles of the 
atomic theory, was proved by implication before the use 
of the balance was introduced into chemistry. Wenzel 
observed, before 1777, that when two neutral substances 
decompose each other, the resulting salts are also neutral. 
In mixing sodium sulphate and barium nitrate, we 
obtain insoluble barium sulphate and neutral sodium 
nitrate. This result could not follow unless the nitric 
acid, requisite to saturate one atom of sodium, were 
exactly equal to that required by one atom of barium, 
so that an exchange could take place without leaving 
either acid or base in excess. 

An important principle of mechanics may also be 
established by a simple acoustical observation. When 
a rod or tongue of metal fixed at one end is set in 
vibration, the pitch of the sound may be observed to 
be exactly the same, whether the vibrations be small or 
great; hence the oscillations are isochronous, or equally 
rapid, independently of their magnitude. On the ground 
of theory, it can be shown that such a result only 
happens when the flexure is proportional to the deflecting 
force. Thus the simple observation that the pitch of 




, 1 


I s 





the sound of a harmonium, for inst^ince, does not chnnge 
with its loudness establishes an exact law of nature.* 

A closely similar instance is found in the proof that the 
intensity of light or heat rays varies inversely as the 
square of the distance increases. For the apparent mag- 
nitude certainly varies according to this law ; hence, if the 
intensity of light varied according to any other law, the 
brightness of an object would be difl'ei*ent at different 
distances, which is not observed to be the case. Melloni 
applied the same kind of reasoning, in a somewhat 
different form, to the radiation of heat-rays. 

Modes of Indirect Measurement, 

Some of the most conspicuously beautiful experiments 
in the whole range of science, have been devised for the 
purpose of indirectly measuring quantities, which in their 
extreme greatness or smallness surpass the powers of 
sense. All that we need to do, is to discover some 
other conveniently measurable phenomenon, which is re- 
lated in a known ratio or according to a known law, 
however complicated, with that to be measured. Having 
once obtained experimentfil data, there is no further 
difficulty beyond that of arithmetic or algebraic calcu- 

Gold is reduced by the gold-beater to leaves so thin, 
that the most powerful microscope would not detect any 
measurable thickness. If we laid several hundred leaves 
upon each other to multiply the thickness, M'e should 
still have no more than ruxf^^ of an inch at the most to 
measure, and the errors arising in the supocposition and 
measurement would be considerable. But we can readily 
obtain an exact result through the connected amount of 
weight. Faraday weighed 2000 leaves of gold, each 
3I inch square, and found them equal to 384 grains. 
From the known specific gravity of gold it was easy to 
calculate that the average thickness of the leaves was 

We must ascribe to Newton the honour of leadinu the 

• Jamin, Coun de Fhynqtie, vol. i. p. 152. 

* Faraday. Chemical Researches, p. y^\. 




way in methods of minute measurement. He did not 
call waves of light by their right name, and did not 
understand their nature; yet he measured their length, 
though it did not exceed the 2,000,000th part of a metre 
or the one fifty- thousandth part of an inch. He pressed 
together two lenses of large but known radii. It was 
easy to calculate the interval between the lenses at any 
point, by measuring the distance from the central point 
of contact. Now, with homogeneous rays the successive 
rings of light and darkness mark the points at which the 
interval between the lenses is equal to one half, or any 
multiple of half a vibration of the light, so that the 
length of the vibration became known. In a similar 
manner many phenomena of interference of rays of light 
admit of the measurement of the wave lengths. Fringes 
of interference arise from rays of light which cross each 
other at a small angle, and an excessively minute dif- 
ference in the lengths of the waves makes a very perceptible 
difference in the position of the point at which two rays 
will interfere and produce darkness. 

Fizeau has recently employed Newton's rings to measure 
small amounts of motion. By merely counting the number 
of rings of sodium monochromatic light passing a certain 
point where two glass plates are in close proximity, he is 
able to ascertain with the greatest accuracy and ease the 
change of distance between these glasses, produced, for 
instance, by the expansion of a metallic bar, connected with 
one of the glass plates.^ 

Nothing excites more admiration than the mode in which 
scientific observers can occasionally measure quantities, 
which seem beyond the bounds of human observation. 
We know the average depth of the Pacific Ocean to be 
14,190 feet, not by actual sounding, which would be 
impracticable in sufficient detail, but by noticing the 
rate of transmission of earthquake waves from the South 
American to the opposite coasts, the rate of movement 
being connected by theory with the depth of the water.' 
In the same way the average depth of the Atlantic Ocean 
is inferred to be no less than 22,157 ^eet, from the velocity 

' Proceedings of the Royal SocxeiVy 30th 
* Herschel, Physical Oeographi/j 1 40. 

November, 1866. 

h ( 


I ' 




of the ordinary tidal waves. A tidal wave again gives 
beautiful evidence of an effect of the law of gravity, 
which we could never in any other way detect. Newton 
estimated that the moon's force in moving the ocean is 
only one part in 2,871400 of the whole force of gravity, 
so that even the pendulum, used with the utmost skill, 
would fail to render it apparent. Yet, the immense extent 
of the ocean allows the accumulation of the effect into a 
very palpable amount ; and from the comparative heights 
of the lunar and solar tides, Newton roughly estimated 
the comparative forces of the moon's and sun's cravitv at 
the earth.i ^ ^ 

A few years ago it might have seemed impossible that 
we should ever measure the velocity with which a star 
approaches or recedes from the earth, since the apparent 
position of the star is thereby unaltered. But the spec- 
troscope now enables us to detect and even measure such 
motions with considerable accuracy, by the alteration which 
it causes in the apparent rapidity of vibration, and conse- 
quently in the refrangibility of rays of Jight of definite 
colour. And while our estimates of the lateral move- 
ments of stars depend upon our very uncertain know- 
ledge of their distances, the spectroscope gives the motions 
of approach and recess irrespective of other motions except- 
ing that of the earth. It gives in short the motions of 
approach and recess of the stars relatively to the earth.* 

The rapidity of vibration for each musical tone, having 
been accurately determined by comparison with the Syren 
(p. 10), we can use sounds as indirect indications of rapid 
vibrations. It is now known that the contraction of a 
muscle arises from the periodical contractions of each 
separate fibre, and from a faint sound or susurrus which 
accompanies the action of a muscle, it is inferred that each 
contraction lasts for about one 300th part of a second 
Minute quantities of radiant heat are now always measured 
indirectly by the electricity which they produce when falJina 
upon a thermopile. The extreme delicacy of the method 
seems to be due to the power of multiplication at several 
points in the apparatus. The number of elements or junc- 

• Prindpiay bk. iii. Prop. 37, Corollaries, 3 tnd 3. Mottc'a 
translation, vol. ii. p. 310. 

* Ro6coe*8 Spectrum Analysis, ist ed. p. 296. 




tions of different metals in the thermopile can be increased 
so that the tension of the electric current derived from the 
same intensity of radiation is multiplied ; the effect of the 
current upon the magnetic needle can be multiplied within 
certain bounds, by passing the current many times round 
it in a coil ; the excursions of the needle can be increased 
by rendering it astatic and increasing the delicacy of its 
suspension ; lastly, the angular divei-gence can be observed, 
with any required accuracy, by the use of an attached 
mirror and distant scale viewed through a telecope (p. 287). 
Such is the delicacy of this method of, measuring heat, that 
Dr. Joule succeeded in making a thermopile which would 
indicate a difference of o°oooi 14 Cent.^ 

A striking case of indirect measurement is furnished by 
the revolving mirror of Wheatstone and Foucault, whereby 
a minute interval of time is estimated in the form of an 
angular deviation. Wheatstone viewed an electric spark 
m a mirror rotating so rapidly, that if the duration of the 
spark had been more than one 72,000th part of a second, 
the point of light would have appeared elongated to an 
angular extent of one-half degree. In the spark, as drawn 
directly from a Leyden jar, no elongation was apparent, so 
that the duration of the spark was immeasurably small ; but 
when the discharge took place through a bad conductor, 
the elongation of the spark denoted a sensible duration.^ 
In the hands of Foucault the rotating mirror gave a 
measure of the time occupied by light in passing through 
a few metres of space. 

Comparative Use of Measuring Instruments, 

In almost every case a measuring instrument serves 
and should serve only as a means of comparison between 
two or more magnitudes. As a general rule, we should 
not attempt to make the divisions of the measuring scale 
exact multiples or submultiples of the unit, but, i-egarding 
them as arbitrary marks, should determine their values by 
companson with the standard itself. The perpendicular 
wu^ in the field of a transit telescope, are fixed at nearly 

! w l^*"^^"^' ^ratM<fij«ton« (1859), vol. cxlix. p. 94. 
• WatU' UxcUonary 0/ ChemiHry, yol ii. p. 393! ^ 

Iv ,' I 








4ii''' ' 




equal but arbitraiy distances, and those distances are afber- 
urards determined, as first suggested by Malvasia, by watch- 
ing the passage of star after star across them, and noting 
the intervals of time by the clock. Owing to the perfectly 
regular motion of the earth, these time intervals give exact 
determinations of the angular intervals. In the same way, 
the angular value of each turn of the screw micrometer 
attached to a telescope, can be easily and accurately 

When a thermopile is used to observe radiant heat, it 
would be almost impossible to calculate on d priori groniidB 
what is the value of each division of the galvanometer 
circle, and still more difficult to constnict a galvanometer, 
so that each division should have a given value. But this 
is quite unnecessary, because by placing the thermopile 
before a body of known dimensions, at a known distance, 
with a known temperature and radiating power, we measure 
a known amount of radiant heat, and inversely measure 
the value of the indications of the thennopile. In a 
similar way Dr. Joule ascertained the actual temperature 
produced by the compression of bars of metal. For having 
inserted a small thermopile composed of a single junction 
of copper and iron wire, and noted the deflections of the 
galvanometer, he had only to dip the bars into water of 
different temperatures, until he produced a like deflec- 
tion, in order to ascertain the temperature developed by 

In some cases we are obliged to accept a very carefully 
constructed instrument as a standard, as in the case of a 
standard barometer or thermometer. But it is then best 
to treat all inferior instruments comparatively only, and 
determine the values of their scales by comparison with 
the assumed standard. 

Systematic Performance of Measurements. 

When a large number of accurate measurements have 
to be effected, it is usually desirable to make a certain 
number of determinations with scrupulous care, and after- 
wards use them as points of reference for the remaining 

' Philotophieal TraiiAfteUont (1859), ^o^- "^ix. p. 119, Ac. 




determinations. In the trigonometrical survey of a coun- 
try, the principal triangulation fixes the relative positions 
and distances of a few points with rigid accuracy. A 
minor triangulation refers every prominent hill or village 
to one of the principal points, and then the detaQs are 
filled in by reference to the secondary points. The survey 
of the heavens is effected in a like manner. The ancient 
astronomers compared the right ascensions of a few prin- 
cipal stars with the moon, and thus ascertained their posi- 
tions with regard to the sun; the minor stars were afterwards 
referred to the principal stars. Tycho followed the same 
method, except that he used the more slowly moving 
planet Venus instead of the moon. Flamsteed was in the 
habit of using about seven stars, favourably situated at 
points all round the heavens. In his early observations 
the distances of the other stars from these standard points 
were determined by the use of the quadrant.^ Even since 
the introduction of the transit telescope and the mural 
circle, tables of standard stars are formed at Greenwich, 
the positions being determined with all possible accuracy, 
so that they can be employed for purposes of reference by 

In ascertaining the specific gravities of substances, all 
gases are referred to atmosj)heric air at a given tempera- 
ture and pressure ; all liquids and solids are referred to 
water. We require to compare the densities of water and 
air with great care, and the comparative densities of any 
two substances whatever can then be ascertained. 

In comparing a very great with a very small magnitude, 
it is usually desirable to break up the process into several 
steps, using intermediate terms of comparison. We should 
never think of measuring the distance from London to 
Edinburgh by laying down measuring rods, throughout the 
whole lengtL A base of- several miles is selected on level 
ground, and compared on the one hand with the standard 
yard, and on the other 'with the distance of London and 
Edinburgh, or any other two points, by trigonometrical 
survey. Again, it would be exceedingly difficult to com- 
pare the light of a star with that of the sun, which would 
be about thirty thousand million times greater ; but Her- 

* Bailj^ Accounl of FlwrnsUed, pp. 378 — 38a 











I :: 





schel ^ effects the comparison by using the full moon as 
an intermediate unit. Wollaston ascertained that the sun 
gave 801,072 times as much light as the full moon, and 
Herschel determined that the light of the latter exceeded 
that of a Centauri 27408 times, so that we find the ratio 
between the light of the sun and star to be that of about 

22,CXX),000,000 to I. 

The Pendulum, 

By far the most perfect and beautiful of all instruments 
of measurement is the pendulum. Consisting merely of a 
heavy body suspended freely at an invariable distance from 
a fixed point, it is most simple in constniction ; yet all the 
highest problems of physical measurement depend upon its 
careful use. Its excessive value arises from two circum- 

(i) The method of repetition is eminently applicable 
to it, as already described (p. 290). 

(2) Unlike other instruments, it connects together three 
different quantities, those of space, time, and force. 

In most works on natural philosophy it is shown, that 
when the oscillations of the pendulum are infinitely small, 
the square of the time occupied by an oscillation is directly 
proportional to the length of the pendulum, and indirectly 
proportional to the force affecting it, of whatever kind. 
The whole theory of the pendulum is contained in the 
formula, first given by Huygens in his Horologium Oscil- 

Time of oscillation = 3*14159 X A/ length of pendulu m 

The quantity 314159 is the constant ratio of the circum- 
ference and radius of a circle, and is of course known with 
accuracy. Hence, any two of the three quantities con- 
cerned being given, the third may be found ; or any two 
being maintained invariable, the third will be invariable. 
Thus a pendulum of invariable length suspended at the 
same place, where the force of gravity may be considered 
constant, furnishes a measure of time. The same invari- 
able pendulum being made to vibrate at different points of 

» Herschel's Aitronomyy ^817, 4th. ed. p. 553 




the earth's surface, and the times of vibration being astro- 
nomically determined, the force of gravity becomes accu- 
rately knowa Finally, with a known force of gravity, 
and time of vibration ascertained by reference to the stars, 
the length is determinate. 

All astronomical observations depend upon the first 
manner of using the pendulum, namely, in the astrono- 
mical clock. In the second employment it has been almost 
equally indispensable. The primary principle that gravity 
IS equal in all matter was proved by Newton's and Gauss' 
pendulum experiments. The torsion pendulum of Michell, 
Cavendish, and Baily, depending upon exactly the same 
pnnciples as the ordinary pendulum, gave the density of 
the earth, one of the foremost natural constants. Kater 
and Sabine, -by pendulum observations in different parts 
of the earth, ascertained the variation of gravity, whence 
comes a determination of the earth's ellipticity. The laws 
of electric and magnetic attraction have also been deter- 
mined by the method of vibrations, which is in constant 
use in the measurement of the horizontal force of terres- 
trial magnetism. 

We must not confuse with the ordinary use of the 
pendulum its application by Neivton, to show the absence 
of internal friction against space,^ or to ascertain the laws 
of motion and elasticity.* In these cases the extent of 
vibration is the quantity measured, and the principles of 
the instrument are different 

Attainable Accuracy of MeasureTTient, 

It is a matter of some interest to compare the degrees 
of accuracy which can be attained in the measurement of 
different kinds of magnitude. Few measurements of any 
kind are exact to more than six significant figures,' but it 
IS seldom that such accuracy can be hoped for. Time is 
the magnitude which seems to be capable of the most exact 
estimation, owing to the properties of the pendulum, and 
the principle of repetition described in previous sections. 

p. 107. 

Prineipia, bt.ii. Sect. 6. Prop. 31. Motte's Translation, vol a 

1 JS^^- ^^' '• ^^ "^- Corollary 6. Motte's Translation, vol. i p. xx 
Thomson and Tait's Natural Philosophy, voL I p. 333. 





i i 


As regards short intervals of time, it has already been 
stated that Sir George Airy was able to estimate one part 
in 8,640,000, an exactness, as he truly remarks, " almost 
beyond conception." ^ The ratio between the mean solar 
and the sidereal day is known to be about one part in 
one hundred millions, or to the eighth place of decimals, 
(p. 289). 

Determinations of weight seem to come next in exact- 
ness, owing to the fact that repetition without error is 
applicable to them. An ordinary good balance should 
show about one part in 500,000 of the load. The finest 
balance employed by M. Stas, turned with one part in 
825,000 of the load.' But balances have certainly been 
constructed to show one part in a million,^ and Ramsden is 
said to have constructed a balance for the Koyal Society, 
to indicate one part in seven millions, though this is hardly 
credible. Professor Clerk Maxwell takes it for granted that 
one part in five millions can be detected, but we ought to 
discriminate between what a balance can do when first 
constructed, and when in continuous use. 

Determinations of length, unless performed with extra- 
ordinary care, are open to much error in the junction of 
the measuring bars. Even in measuring the base line of 
a trigonometrical survey, the accuracy generally attained 
is only that of about one part in 60,000, or an inch in the 
mDe; but it is said that in four measurements of a 
base line carried out very recently at Cape Comorin, the 
greatest error was 0*077 ^^^^ ^^ ^ '^^ mile, or one part in 
1 ,382,400, an almost incredible degree of accuracy. Sir J. 
Whitworth has shown that touch is even ap-more delicate 
mode of measuring lengths than sight, and by means of a 
splendidly executed screw, and a small cube of ii*on placed 
between two flat-ended iron bars, so as to be suspended 
when touching them, he can detect a change of dimension 
in a bar, amounting to no more than one-millionth of ai: 

* Philosophical Trtmsactioru, (1856), vol. cxlvi. pp. 330, 331. 
^ First Annual Report of the Minty p. 106. 

3 Jevons, ill Watts' Dictionary of Chemistry ^ vol. i. b. 483. 

* British Association, Glasgow, 1 856. Address of we Preside^it of 
ike MeehanictU Section. 



As we have seen, instruments of measurement are 
only means of comparison between one magnitude and 
another, and as a general rule we must assume some 
one arbitrary magnitude, in terms of which all results 
of measurement are to be expressed. Mere ratios be- 
tween any series of objects will never tell us their 
absolute magnitudes ; we must have at least one ratio 
for each, and we must have one absolute magnitude. The 
number of ratios n are expressible in n equations, which 
will contain at least n + 1 quantities, so that if we 
employ them to make known n magnitudes, we must 
have one magnitude known. Hence, whether we are 
measuring time, space, density, mass, weight, energy, or 
any other physical quantity, we must refer to some con- 
ci-ete standard, some actual object, which if onc^ lost and 
irrecoverable, all our measures lose their absolute mean- 
ing. This concrete standard is in all cases arbitrary in 
point of theory, and its selection a question of practical 

There are two kinds of magnitude, indeed, which do not 
need to be expressed in terms of arbitrary concrete units, 
since they pre-suppose the existence of natural standard 
umts. One case is that of abstract number itself, which 
needs no special unit, because any object which exists or 
IS thought of as separate fi-om other objects (p. 157) fur- 
nishes us with a unit, and is the only standard required. 

Angular magnitude is the second case in which 
we have a natural unit of referenci?, namely the wholo 








revolution or perigon, as it has been called by Mr. Sande- 
man.^ It is a necessary result of the uniform properties 
of space, that all complete revolutions are equal to each 
other, so that we need not select any one revolution, but 
can always refer anew to space itself. Whether we take 
the whole perigon, its half, or its quarter, is really imma- 
terial ; Euclid took the right angle, because the Greek geo- 
meters had never generalised their notions of angular 
magnitude sufficiently to treat angles of all magnitudes, or 
of unlimited qxtaiUity of revolution. Euclid defines a right 
angle as half that made by a line with its own continuation, 
which is of course equal to half a revolution, but which 
was not treated as an angle by him. In mathematical 
analysis a different fraction of the perigon is taken, namely, 
such a fraction that the arc or portion of the circumference 
included within it is. equal to the radius of the circle. In 
this point of view angidar magnitude is an abstract ratio, 
namely, the ratio between the length of arc subtended and 
the length of the radius. The geometrical unit is then 
necessarily the angle corresponding to the ratio unity. 
This angle is equal to about 57^ 17', 44" '8, or decimally 
57°-2957795 13... .* It was called by De Morgan the araial 
unit, but a more convenient name for common use would 
be radian, as suggested by Professor Everett. Though this 
standard angle is naturally employed in mathematical 
analysis, and any other unit would introduce great com- 
plexity, we must not look upon it as a distinct imit, since 
its amount is connected with that of the half perigon, 
by the natural constant 3*14159 . . . usutdly denoted by 
the letter ir. 

When we pass to other species of quantity, the choice 
of unit is found to be entirely arbitrary. Thei-e is abso- 
lutely no mode of defining a length, but by selecting some 
physical object exhibiting that length between certain 
obvious points — as, for instance, the extremities of a bar, 
or marks made upon its surface. 

' Pdicoietics, or the Science of Quantity ; an Elementary Treaiiu on 
Algebra, and iU groundwork Arithmetic. By Archiball Saiuleinan, 
M.A. Cambridfje (Deighton, Bel], and Co.), 1868, p. 304. 

* De Morgan's Trigonometry and DouhU Algebra, p. 5. 


Standard Unit of Time. 

Time is the great independent variable of all change 

that which itself flows on uninterruptedly, and brings the 
variety which we call motion and life. When we reflect 
upon its intimate nature. Time, like every other element of 
existence, proves to be an inscrutable mystery. We can 
only say with St. Augustin, to one who asks us what is 
time, "I know when you do not ask me." The mind of 
man will ask what can never be answered, but one result 
of a true and rigorous logical philosophy must be to 
convince us that scientific explanation can only take place V 
between phenomena which have something in common, 
and that when we get down to primary notions, like those 
of time and space, the mind must meet a point of mystery 
beyond which it cannot penetrate. A definition of time 
must not be looked for ; if we say with Hobbes,* that it 
is '* the phantasm of before and after in motion," or with 
Aristotle that it is " the number of motion according to 
former and latter," we obviously gain nothing, because 
the notion of time is involved in the expressions hefore 
and after, former and latter. Time is undoubtedly one 
of those primary notions which can only l)e defined physi- 
cally, or by observation of phenomena which proceed in 

If we have not advanced a step beyond Augustin's acute 
reflections on this subject,* it is curious to observe the 
wonderful advances which have been made in the practical 
measurement of its efllux. In earlier centuries the rude 
sun-dial or the rising of a conspicuous star gave points of 
reference, while the flow of water from the clepsydra, the 
burning of a candle, or, in the monastic ages, even' the 
continuous chanting of psalms, were the means of roughly 
subdividing periods, and marking the hours of the day and 
mgbt.8 The sun and stars still furnish the standard of 
time, but means of accurate subdivision have become 
requisite, and this has been furnished by the pendulum 

\ ^'^l^. ^^^^ «/ ^r^«- ffobbes, Edit, by Molesworth, vol. i. p. qc 
^ Confesstons, bk. xi. chapters 20—28. 

«,^ ^* ^* ^f^ ^^^ ^^y curious particulars conceniine the 
measurement of time In hk AHronomy 0/ the Ancients, pp. 24i,^&a 

X 2 






[oh A p. 

and the chronograpli. By the pendulum we can accurately 
divide the day into seconds of time. By the chronograph 
we can subdivide the second into a hundred, a thousand, 
or even a million parts. "Wheats tone measured the dura- 
tion of an electric spark, and found it to be no more than 
one 115,200th part of a second, while more recently 
Captain Noble has been able to appreciate intervals of 
time not exceeding the millionth part of a second. 

When we come to inquire precisely what phenomenon 
it is tliat we thus so minutely measure, we meet insur- 
mountable difficulties. Newton distinguished time accord- 
ing as it was absolute or apparent time, in the following 
words : — " Absolute, true, and mathematical time, of itself 
and from its own nature, flows equably without regard to 
anything external, and by another name is called duration; 
relative, apparent and common time, is some sensible and 
external measure of duration by the means of motion."* 
Though we are perhaps obliged to assume the existence 
of a uniformly increasing quantity which we call time, 
yet we cannot feel or know abstract and absolute time. 
Duration must be made manifest to us by the recurrence 
of some phenomenoa The succession of our own thoughts 
is no doubt the first and simplest measure of time, but a 
very rude one, because in some persons and circumstances 
the thoughts evidently flow with much greater rapidity 
than in other persons and circumstances. In the absence 
of all other phenomena, the interval between one thought 
and another would necessarily become the unit of time, 
but the most cursory observations show "that there are 
changes in the outward world much better fitted by their 
constancy to measure time than the change of thoughts 
within us. 

The earth, as I have already said, is the real clock of tlie 
astronomer, and is practically assumed as invariable in 
its movements. But on what ground is it so assumed? 
According to the first law of motion, every body perseveres 
in its state of rest or of uniform motion in a right line, 
unless it is compelled to change that state by forces im- 
pressed thereon. Eotatory motion is subject to a like 

^ FrineipiOf bk. i. Scholium io D^niliont. Timnfllated by Motta^ 
wn\. \ p. a Sec alio p. 1 1. 


condition, namely, that it perseveres uniformly unless dis- 
turbed by extrinsic forces. Now uniform motion means 
motion through equal spaces in equal times, so that if we 
have a body entirely free from all resistance or perturba- 
tion, and can measure equal spaces of its path, we have a 
perfect measure of time. But let it be remembered that 
this law has never been absolutely proved by experience • 
for we cannot point to any body, and say that it is wholly 
unresisted or undisturbed ; and even if we had such a body 
we should need some independent standard of time to 
ascerUm whether its motion was really uniform As it 
IS m movmg bodies that we find the best standard of time 
we cannot use them to prove the uniformity of their own 
movements, which would amount to a petitio priudpii. 
Our experience comes to this, that when we examine and 
compare the movements of bodies which seem to us nearly 
free from disturbance, we find them giving nearly har- 
monious measures of time. If any one body which seems 
U, us to move umformly is not doing so, but is subject to 
hts and starts unknown to us, because we have no absolute 
standard of time, then all other bodies must be subject to 
the same arbitrary fits and starts, otherwise there would be 
discrepancy disclosing the irregularities. Just as in com- 
paring together a number of chronometers, we should soon 

u^^Ik .1,''''*'' ^^ *^^^' ^^^^- irregularly, as compared 
with the others, so m nature we detect disturbed movement 
by ite discrepancy from that of other bodies which we 
tjBUeve to be undisturbed, and which agree nearly amonc^ 
thenaselves. But inasmuch as the measure ot motioS 
involves time, and the measure of time involves motion, 
there must be ultimately an assumptioa We may define 
equal times, as times during which a moving body under 
the influence of no force describes equal spaces ; ^ but all 
we can say m support of this definition is, that it leads us 
into no known difficulties, and that to the best of our ex- 
perience one freely moving body gives the same results as 
any other. 

When we inquire where the freely moving body is. no 
perfectly satisfactory answer can be givea Practically 
the rotating globe is sufficiently accurate, and Thomson 

» Rankine, Fhilosophieal Magazitu, Feb. 1867, voL xxxiu p. 91. 











and Tait say: "Equal times are times during which the 
earth turns through equal angles."* No long time has 
passed since astronomers thought it impossible to detect 
any inequality in its movement. Poisson was supposed 
to have proved that a change in th© length of the sidereal 
day amounting to one ten-millionth part in 2,500 years was 
incompatible with an ancient eclipse recorded by the 
Chaldaeans, and similar calculations were made by Laplace. 
But it is now known that these calculations were some- 
what in error, and that the dissipation of enei-gy arising 
out of the friction of tidal waves, and the radiation of the 
heat into space, has slightly decrcased the rapidity of the 
earth's rotatory motion. The sidereal day is now longer by 
one part in 2,700,000, than it was in 720 B.C. Even l^efoi-e 
this discovery, it was known that invariability of rotation 
depended upon the perfect maintenance of the earth's 
internal heat, which is requisite in order that the earth's 
dimensions shall be unaltered. Now the earth being 
superior in temperature to empty space, must cool more or 
less rapidly, so that it cannot furnish an absolute measure 
of time. Similar objections could be raised to all other 
rotating bodies within our cognisance. 

The moon's motion round the earth, and the earth's 
motion round the sun, form the next best measure of 
time. They are subject, indeed, to disturbance from other 
planets, but it is believed that these perturbations must 
in the course of time run through their rhythmical courses, 
leaving the mean distances unafi'ected, an<k consequently, 
by the third Law of Kepler, the periodic times unchanged. 
But there is more reason than not to believe that the earth 
encounters a slight resistance in passing through space, 
like that which is so apparent in Encke's comet There 
may also be dissipation of energy in the electrical relations 
of the earth to the sun, possibly identical with that which 
is manifested in the retardation of comets.* It is probably 
an untrue assumption then, that the earth's orbit remains 
quite invariable. It is just possible that some other body 
may be found in the course of time to furnish a better 

* Treatise 011 Natural Philosophy , vol. i. p. 179. 

• Proceedings of the ManehetUr Philosophical ISocieiyf 28tb Not. 
1871, vol xi. p. 33. 


standard of time than the earth in its annual motion. 
The greatly superior mass of Jupiter and its satellites, and 
their greater distance from the sun, may render the 
electrical dissipation of energy less considerable than in 
the case of the earth. But the choice of the best measure 
will always be an open one, and whatever moving body 
we choose may ultimately be shown to be subject to 
disturbing forces. 

The pendulum, although so admirable an instrument for 
subdivision of time, fails as a standard ; foi though the 
same pendulum affected by the same force of gravity per- 
forms equal vibrations in equal times, yet the slightest 
change in the form or weight of the pendulum, the least 
corrosion of any part, or the most minute displacement of 
the point of suspension, falsifies the results, and there enter 
many other difficult questions of temperature, friction, 
resistimce, length of vibration, &c. 

Thomson and Tait are of opinion * that the ultimate 
standard of chronometry must be founded on the physical 
properties of some body of more constant character than 
the earth ; for instance, a carefully arranged metallic 
spring, hermetically sealed in an exhausted glass vessel. 
But it is hard to see how we can be sure that the dimen- 
sions and elasticity of a piece of wrought metal will 
remain perfectly unchanged for the few millions of years 
contemplated by them. A nearly perfect gas, like 
hydrogen, is perhaps the only kind of substance in the 
unchanged elasticity of wliich we could have confidence. 
Moreover, it is difl&cult to perceive how the undulations of 
such a spring could be observed with the requisite 
accuracy. Mere recently Professor Clerk Maxwell has 
made the novel suggestion, discussed in a subsequent 
section, that undulations of light in vacuo would form the 
most univei-sal standard of reference, both as regards time 
and space. According to this system the unit of time 
would be the time occupied by one vibration of the par- 
ticular kind of light whose wave length is taken as the 
unit of length. 

* Th^ JSUmeiits of Nahtrai Philosophy, part i. p. 119. 


'I • 










The Unit of Space and the Bar Standard, 

Next in importance after the measurement of time is 
/y that of space. Time comes first in theory, because pheno- 
^ mena, our internal thoughts for instance, may change in 
time without regaixi to space. As to the phenomena 
of outward nature, they tend more and more to resolve 
themselves into motions of molecules, and motion cannot 
be conceived or measured without reference both to time 
and space. 

Turning now to space measurement, we find it almost 
equally difficult to fix and define once and for ever, a unit 
magnitude. There are three different modes in which 
it has been proposed to attempt the perpetuation of a 
standard length. 

(i) By constructing an actual specimen of the standard 
yard or metre, in the form of a bar. 

(2) By assuming the globe itself to be the ultimate 
standard of magnitude, the practical unit being a sub- 
multiple of some dimension of the globe. 

(3) By adopting the length of the simple seconds pen- 
dulum, as a standard of reference. 

At first sight it might seem that there was no great 
difficulty in this matter, and that any one of these methods 
might serve well enough; but the more minutely we 
inquire into the details, the more hopeless appears to be 
the attempt to establish an invariable standard. We must 
in the first place point out a principle not^f an obvious 
character, namely, that the standard length must be defined 
hy one single object} To make two bars of exactly the 
sanie length, or even two bars bearing a perfectly defined 
ratio to each other, is beyond the power of human art If 
two copies of the standard metre be made and declared 
equally correct, future investigators will certainly discover 
some discrepancy between them, proving of course that they 
cannot both be the standard, and giving cause for dispute 
as to what magnitude should then be taken as correct. 

If one invariable bar could be constructed and main- 
tained as the absolute standard, no such inconvenience 
could arise. Each successive generation as it acquired 

« See Hams' i7«My ttpon Money and Coins, part ii. [1758] p. 127. 


higher powers of measurement, would detect errors in 
the copies of the standard, but the standard itself would 
be unimpeached, and would, as it were, become by degrees 
more and more accurately known. Unfortunately to con- 
struct and preserve a metre or yard is also a task which 
is eitlier impossible, or what comes nearly to the same 
thing, cannot be shown to be possible. Passing over the 
practical difficulty of defining the ends of the standard 
length with complete accuracy, whether by dots or lines 
on the surface, or by the terminal points of the bar, we 
have no means of proving that substances remain of in- 
variable dimensions. Just as we cannot tell whether the 
rotation of the earth is uniform, except by comparing it 
with other moving bodies, believed to be more uniform 
in motion, so we cannot detect the change of length in a 
bar, except by comparing it with some other bar sup- 
posed to be invariabla But how are we to know which 
is the invariable bar? It is certain that many rigid 
and apparently invariable substances do change in di- 
mensions. The bulb of a thermometer certainly contracts 
by age, besides undergoing rapid changes of dimensions 
when wanned or cooled through 100** Cent. Can we 
be sure that even the most solid metallic bars do not 
slightly contract by age, or undei-go variations in their 
structure by change of temperature. Fizeau was induced 
to try whether a quartz crystal, subjected to several 
hundi'ed alternations of temperature, would be modified in 
its physical properties, and he was unable to detect any 
change in the coefficient of expansion.^ It does not 
follow, however, that, because no apparent change was 
discovered in a quartz crystal, newly-construct-ed bars of 
metal would undergo no change. 

The best principle, as it seems to me, upon which the 
^rpetuation of a standard of length can be rested, is that, 
if a variation of length occurs, it will in all probability be 
of different amount in different substances. If then a 
great number of standard metres were constructed of all 
kmds of different metals and alloys ; hard rocks, such as 
granite, serpentine, slate, quartz, limestone; artificial 
substances, such as porcelain, glass, &c., &c., careful 

' Philoiophical Maganne, (1868), 4th Series, yoL xxxvi. p. 32. 










comparison would show from time to time the comparative 
variations of length of these different substances. The 
most variable substances would be the most divergent, and 
the standard would be furnished by the mean length 
of those which agreed most closely with each other just 
as uniform motion is that of those bodies which agree 
most closely in indicating the efflux of time. 

Th>e Terrestrial Standard. 

The second method assumes that the globe itself is a 
body of invariable dimensions and the founders of the me- 
trical system selected the ten- millionth part of the dis- 
tance from the equator to the pole as the definition of the 
metre. The first imperfection in such a method is that the 
earth is certainly not invariable in size; for we know 
that it is superior in temperature to surrounding space, and 
must be slowly cooling and contracting. There is much 
reason to believe that all earthquakes, volcanoes, mountain 
elevations, and changes of sea level are evidences of this 
contraction as asserted by Mr. Mallet.^ But such is the 
vast bulk of the earth and the duration of its past exis- 
tence, that this contraction is perhaps less rapid in propor- 
tion than that of any bar or other material standard which 
we can construct. 

The second and chief difficulty of this method arises 
from the vast size of the earth, which prevents us from 
making any comparison with the ultiifiate standard, ex- 
cept by a trigonometrical survey of a most elaborate and 
costly kind. The French physicists, who first proposed 
the method, attempted to obviate this inconvenience by 
carrying out the survey once for all, and then constructing 
a standard metre, which should be exactly the one ten 
millionth part of the distance from the pole to the 
equator. But since all measuring operations are merely 
approximate, it was impossible that this operation could be 
perfectly achieved. Accordingly, it was shown in 1838 
that the supposed French metre was erroneous to the con- 
siderable extent of one part in 5527. It then became 
necessary either to alter the length of the assumed metre, 

* Froe4sedituu of the Boval Society, ?oth June, 1872, rq\, xx. p. 438, 


or to abandon its supposed relation to the earth's dimen- 
sions. The French Government and the International 
Metrical Commission have for obvious reasons decided in 
favour of the latter course, and have thus reverted to the 
first method of defining the metre by a given bar. As 
from time to time the ratio between this assumed standard 
metre and the quadrant of the earth becomes more accu- 
rately known, we have better means of restoring that metre 
by reference to the globe if required. But until lost, des- 
troyed, or for some clear reason discredited, the bar metre 
and not the globe is the standard. Thomson and Tait re- 
mark that any of the more accurate measurements of the 
English trigonometrical survey might in like manner be 
employed to restore our standard yard, in terms of which 
the results are recorded. 

The Pendulum Standard. 

The third method of defining a standard length, by 
reference to the seconds pendulum, was first proposed by 
Huyghens, and was at one time adopted by the English 
Government. From the principle of the pendulum (p. 302) 
it clearly appears that if the time of oscillation and the 
force actuating the pendulum be the same, the length of 
the pendulum must be the same. We do not get rid of 
theoretical difficulties, for we must assume the attraction 
of gravity at some point of the earth's surface, say 
London, to be unchanged from time to time, and the 
sidereal day to l)e invariable, neither assumption being 
absolutely correct so far as we can judge. The pendulum, 
in short, is only an indirect means of making one physical 
quantity of space depend upon two other physical quan- 
tities of time and force; 

The practical difficulties are, however, of a far more 
serious character than the theoretical ones. The length 
of a pendulum is not the ordinary length of the instru- 
ment, which might be greatly varied without affecting the 
duration of a vibration, but the distance from the centre of 
suspension to the centre of oscillation. There are no 
direct means of determining this latter centre, which 
depend* upon the average momentum of all the particles 













of the pendtdum as regards the centre of suspension. 
Huyghens discovered that the centres of suspension 
and oscillation are interchangeable, and Kater pointed out 
that if a pendulum vibrates with exactly the same rapidity 
when suspended from two different points, the distance 
between these points is the true length of the equivalent 
simple pendulum.^ But the practical difficulties in em- 
ploying Kater's reversible pendulum are considerable, and 
questions regarding the disturbance of the air, the force 
of gravity, or even the interference of electrical attractions 
have to be entertained. It has been shown that all the 
experiments made under the authority of Government for 
determining the ratio between the standard yard and the 
seconds pendulum, were vitiated by an error in the correc- 
tions for the resisting, adherent, or buoyant power of the 
air in which the pendulums were swung. Even if such 
corrections were rendered unnecessary by opemting in a 
vacuum, other difficult questions remain.* Gauss' mode of 
comparing the vibrations of a wire pendulum when sus- 
pended at two different lengths is open to equal or greater 

practical difficulties. Thus it is found that the pendulum 
standard cannot compete in accuracy and certainty with 
the simple bar standard, and the method would only be 
useful as an accessory mode of restoring the bar standard 
if at any time again destroyed. 

Unit of Density, / 

Before we can measure the phenomena of nature, we 
require a third independent unit, which shall enable us to 
define the quantity of matter occupying any given space. 
All the changes of nature, as we shall see, are probably so 
many manifestations of energy ; but energy requires some 
substratum or material machinery of molecules, in and by 
which it may be manifested. Observation shows that, as 
regards force, there may be two modes of variation of 
matter. As Newton says in the first definition of the 
Principia, ** the quantity of matter is the measure of the 
same, arising from its density and bulk conjunctly.** 

1 Kater*8 Treatise (m Mechanics, Cabinet Cyclopadia, p. 154. 
* Grant's History 0/ Physical Astronomy, p. 156. 


Thus the force required to set a body in motion varies 
both according to the bulk of the matter, and also accord- 
ing to its quality. Two cubic inches of iron of uniform 
quality, will require twice as much force as one cubic inch 
to produce a certain velocity in a given time ; but one cubic 
inch of gold will require more force than one cubic inch of 
iron. There is then some new measurable quality in 
matter apart from its bulk, which we may call density, and 
which is, strictly speaking, indicated by its capacity to 
resist and absorb the action of force. For the unit of 
density we may assume that of any substance which is uni- 
form in quality, and can readily be referred to from time to 
time. Pure water at any definite temperature, for instance 
that of snow melting under inappreciable pressure, fur- 
nishes an invariable standard of density, and by compar- 
ing equal bulks of various substances with a like bulk of 
ice-cold water, as regards the velocity produced in a unit 
of time by the same force, we should ascertain the densities 
of those substances as expressed in that of water. Practi- 
cally the force of gravity is used to measure density ; for a 
beautiful experiment with the pendulum, performed by 
Newton and repeated by Gauss, shows that all kinds of 
matter gravitate equally. Two portions of matter then 
which are in equilibrium in the balance, may be assumed 
to possess equal inertia, and their densities will therefore 
be inversely as their cubic dimensions. 

Unit of Mass, 

Multiplying the number of units of density of a portion 
of matter, by the number of units of space occupied by it, 
we arrive at the quantity of matter, or, as it is usually 
called, the unit of mass, as indicated by the inertia and 
gravity it possesses. - To proceed in the most simple 
manner, the unit of mass ought to be that of a cubic unit 
of matter of the standard density ; but the founders of 
the metrical system took as their unit of mass, the cubic 
centimetre of water, at the temperature of maximum 
density (about 4* Cent). They ceiled this unit of mass 
the gramme, and constructed standard specimens of the 
kilogram, which might be readily referred to by all who 
required to employ accurate weights. Unfortunately the 




determination of the bulk of a given weight of water at a 
certain temperature is an operation involving many dif 
ficulties, and it cannot be performed in the present day 
with a greater exactness than that of about one part in 
SOOOt the results of careful observers being sometimes 
found to differ as much as one part in locx).* 

Weights, on the other hand, can be compared with 
each other to at least one part in a million. Hence if 
different specimens of the kilogram be prepared by direct 
weighing against water, they will not agree closely with 
each other ; the two principal standard kilograms agree 
neither with each other, nor with their definition. Accord- 
ing to Professor Miller the so-called Kilogramme des 
Archives weighs 15432-34874 grains, while the kilogram 
deposited at the Ministry of the Interior in Paris, as the 
standard for commercial purposes, weighs 1 5432*344 grains. 
Since a standard weight constructed of platinum, or plati- 
num and iridium, can be preserved free from any appreci- 
able alteration, and since it can be very accurately com- 
pared with other weights, we shaU ultimately attain the 
greatest exactness in our measurements of mass, by assum- 
ing some single kilogram as a provisional standard, leaving 
the determination of its actual mass in units of space and 
density for future investigation. This is what is practi- 
cally done at the present day, and thus a unit of mass 
takes the place of the unit of density, both in the French 
and English systems. The English pound is defined by a 
certain lump of platinum, preserved &t Westminster, and 
is an arbitrary mass, chosen merely that it may agree as 
nearly as possible with old English pounds. The gallon, 
the old English unit of cubic measurement, is defined by 
the condition that it shall contain exactly ten pounds 
weight of water at 62* Fahr. ; and although it is stated that 
it has the capacity of about 277*274 cubic inches, this 
ratio between the cubic and linear systems of measure- 
ment is not legally enacted, but left open to investigation. 
While the French metric system as originally designed 
was theoretically perfect, it does not differ practically in 
this point from the English system. 

> Olerk Maxwell's Theory of Heai^ p. 79. 


Natural System of Standards. 

Quite recently Professor Clerk Maxwell has suggested 
that the vibrations of light and the atoms of matter might 
conceivably be employed as the ultimate standards of 
length, time, and mass. We should thus arrive at a 
natural system of standards, which, though possessing no 
present practical importance, has considerable theoretical 
interest " In the present state of science," he says, " the 
most universal standai'd of length which we could assume 
would be the wave-length in vacuum of a particular kind 
of light, emitted by some widely diffused substance such 
as sodium, which has well-defined lines in its spectrum. 
Such a standard would be independent of any changes in 
the dimensions of the earth, and should be adopted by 
those who expect their writings to be more permanent than 
that body." * In the same way we should get a universal 
standard unit of time, independent of all questions about 
the motion of material bodies, by taking as the unit the 
periodic time of vibration of that particular kind of light 
whose wave-length is the unit of length. It would follow 
that with these units of length and time the unit of 
velocity would coincide with the velocity of light in empty 
space. As regards the unit of mass. Professor Maxwell, 
humorously as I should think, remarks that if we expect 
soon to be able to determine the mass of a single molecule 
of some standard substance, we may wait for this deter- 
mination before fixing a universal standard of mass. 

In a* theoretical point of view there can be no reasonable 
doubt that vibrations of light are, as far as we can tell, the 
most fixed in magnitude of all phenomena. There is as 
usual no certainty in the matter, for the properties of the 
basis of light may vary to some extent in different parts of 
space. But no differences could ever be established in the 
velocity of light in different parts of the solar system, and 
the spectra of the stars show that the times of vibration 
there do not differ perceptibly from those in this part of 
the universe. Thus all presumption is in favour of the 
absolute constancy of the vibrations of light — absolute, 
that is, so far as regards any means of investigation we are 

* TrwUu on Electricity cuid Magnetiemy toL i. p. 3> 








likely to possess. Nearly the same considerations apply 
to the atomic weight as the standard of mass. It is im- 
possible to prove that all atoms of the same substance are 
of equal mass, and some physicists think that they differ, so 
that the fixity of combining proportions may bo due only 
to the approximate constancy of the mean of countless 
millions of discrepant weights. But m any case the do 
tection of difference is probably beyond our powers. In a 
theoretical point of view, then, the magnitudes suggested 
by Professor Maxwell seem to be the most fixed ones of 
wliich we have any knowledge, so that they necessarily 
become the natui-al units. 

In a practical point of view, as Professor Maxwell would 
be the first to point out, they are of little or no value, be- 
cause in the present state of science we caimot measure a 
vibration or weigh an atom with any approach to the 
accuracy which is attainable in the comparison of standard 
metres and kilograms. The velocity of light is not known 
probably within a thousandth part, and as we progress in 
the knowledge of light, so we shall progress in the accu- 
rate fixation of other standards. All that cau be said then, 
is that it is very desirable to determine the wave-lengths 
and periods of the principal lines of the solar spectrum, 
and the absolute atomic weights of the elements, with all 
attainable accuracy, in terms of our existing standards. 
The numbers thus obtained would^admit of the reproduc- 
tion of our standards in some future age of the world lo a 
corresponding degree of accuracy, were there need of such 
reference ; but so far as we can see at present, there is no 
considerable probability that this mode of repi*oduction 
would ever be the best mode. 

Subsidiary Units, 

Having once established the standard units of time, 
space, and density or mass, we might employ them for the 
expression of all quantities of such nature. But it is often 
convenient in particular branches of science to use mul- 
tiples or submultiples of the original units, for the ex- 
pression of quantities in a simple manner. We use the 
mile rather than the yard when treating of the magnitude 
of the globe, and the mean distance of the earth and 


sun is not too large a unit when we have to describe 
the distances of the stars. On tlie other hand, when we 
are occupied with microscopic objects, the inch, the line 
or the millimetre, become the most convenient terms of 

It is allowable for a scientific man to introduce a new 
unit in any branch of knowledge, provided that it assists 
precise expression, and is carefully brought into relation 
with the primary units. Thus Professor A. W. Williamson 
has proposed as a convenient unit of volume in chemical 
science, an absolute volume equal to about 11*2 litres 
representing the bulk of one gram of hydrogen gas at 
standard temperature and pressure, or the equivalent weight 
of any other gas, such as 16 grams of oxygen, 14 grams 
of nitrogen, &c. ; in short, the bulk of that quantity of 
any one of those gases which weighs as many grams as 
there are units in the number expressing its atomic 
weight.^ Hofmann has proposed a new unit of weight for 
chemists, called a crith, to be defined by the weight of one 
litre of hydrogen gas at 0° C. and o°76 mm., weighing 
about 0*0896 gram.* Both of these units must be re- 
garded as purely subordinate units, ultimately defined by 
reference to the primary units, and not involving any new 

Derived Units, 

The standard units of time, space, and mass having been 
once fixed, many kinds of magnitude are naturally measured 
by units derived from them. From the metre, the unit of 
linear magnitude follows in the most obvious manner the 
centiare or square metre, the unit of superficial magnitude, 
and the litre that is the cube of the tenth part of a metre, 
the unit of capacity or volume. Velocity of motion is ex- 
pressed by the ratio of the space passed over, when the 
motion is uniform, to the time occupied ; hence the unit 
of velocity is that of a body which passes over a unit 
of space in a unit of time. In physical science the 
unit of velocity might be taken as one metre per second. 

^Chemistry for Stv.dents, by A. W. Williamson. Clarendon Press 
Senes, 2nd ed. Preface p. vi. * Introduction to Chemistry, p. 131. 









Momentum is measured by the mass moving, regard being 
paid both to the amount of matter and the velocity at 
which it is moving. Hence the unit of momentum will be 
that of a unit volume of matter of the unit density moving 
with the unit velocity, or in the French system, a cubic 
centimetre of water of the maximum density moving one 
metre per second. 

An accelerating force is measured by the ratio of the 
momentum generated to the time occupied, the force 
being supposed to act uniformly. The unit of force will 
therefore be that which generates a unit of momentum 
in a unit of time, or which causes, in the French system, 
one cubic centimetre of water at maximum density to 
acquire in one second a velocity of one metre per second. 
The force of gravity is the most familiar kind of force, 
and as, when acting unimpeded upon any substance, it 
produces in a second a velocity of 9-80868 . . metres 
per second in Paris, it follows that the absolute unit 
of force is about the tenth part of the force of gravity. 
If we employ British weights and measures, the absolute 
unit of force is represented by the gravity of about half 
an ounce, since the force of gravity of any portion of 
matter acting upon that matter during one second, pro- 
duces a final velocity of 32* 1889 feet per second or about. 
32 units of velocity. AlthouglTfrom its perpetual action 
and approximate uniformity we find in gravity the most 
convenient force for reference, and thus habitually employ 
it to estimate quantities of matter, we must remember 
that it is only one of many instances of force. Strictly 
speaking, we should express weight in terms of force, but 
practically we express other forces in terms of weight 

We still require the unit of energy, a more com- 
plex notion. The momentum of a body expresses the 
quantity of motion which belongs or would belong to the 
aggregate of the particles ; but when we consider how this 
motion is related to the action of a force producing or 
removing it, we find that the effect of a force is pro- 
portional to the mass multiplied by the square of the 
velocity and it is convenient to take half this product 
as the expression required. But it is shown in books 
upon dynamics that it will be exactly the same thing if 
we define energy by a force acting through a space. The 


natural unit of energy will then be that which overcomes 
a unit of force acting through a unit of space; when we 
lift one kilogram through one metre, against gravity, we 
therefore accomplish 9-80868 . . units of work, that is,' we 
lum so many units of potential energy existing in the 
muscles, into potential energy of gravitation. In liftin^r 
one pound through one foot there is in like manner a con- 
version of 32- 1 889 units of energy. Accordingly the 
unit of energy will be in the English system, that required 
to lift one pound through about the thirty-second part of 
a foot; in terms of metric units, it will be that required to 
lift a kilogram through about one tenth part of a metre. 

Every person is at liberty to measure and record 
quantities in terms of any unit which he likes. He 
may use the yard for linear measurement and the litre 
for cubic measurement, only there will then be a com- 
plicated relation between his different results. The 
system of derived units which we have been briefly con- 
sidering, is that which gives the most simple and natural 
relations between quantitative expressions of different 
kinds, and therefore conduces to ease of comprehension 
and saving of laborious calculation. 

It would evidently be a source of great convenience if 
scientific men could agree upon some single system of 
units, original and derived, in terms of which all quantities 
could be expressed. Statements would thus be rendered 
easily comparable, a large part of scientific literature would 
be made intelligible to all, and the saving of mental labour 
would be immense. It seems to be generally allowed, too, 
that the metric system of. weights and measures presents 
the best basis for the ultimate system; it is thoroughly 
established in Western Europe ; it is legalised in England ; 
it IS already commonly employed by scientific men; it is 
m Itself the most simple and scientific of systems. There 
IS every reason then why the metric system should be 
accepted at least in its main features. 

Jhrovisiorud Units, 

Ultimately, a« we can hardly doubt, all phenomena 
will be recognised as so many manifestations of energy: 
and, being expressed in terms of the unit of energy, wUl 

Y 2 






be referable to the primary units of space, time, and 
density. To effect this reduction, however, in any pwrticu- 
lar case, we must not only be able to compare different 
quantities of the phenomenon, but to trace the whole 
series of steps by which it is connected with the primary 
notions. We can readily observe that the intensity of 
one source of light is greater than that of another ; and, 
knowing that the intensity of light decreases as the 
square of the distance increases, we can easily determine 
their comparative brilliance. Hence we can express the 
intensity of light falling upon any surface, if we have a 
unit in which to make the expression. Light is un- 
doubtedly one form of energy, and the unit ought therefore 
to be the unit of energy. But at present it is quite im- 
possible to say how much energy there is in any particular 
amount of light. The question then arises, — Are we to 
defer the measurement of light until we can assign its 
relation to other forms of energy ? If we answer Yes, it is 
equivalent to saying that the science of light must stand 
still perhaps for a generation ; and not only this science 
but many others. The true course evidently is to select, 
as the provisional unit of light, some light of convenient 
intensity, which can be reproduced from time to time in 
the same intensity, and which is-defiued by physical cir- 
cumstances. All the phenomena of light may be experi- 
mentally investigated relatively to this unit, for instance 
that obtained after much labour by Bunsen and Eoscoe.^ 
In after years it will become a matter of inquiry what is 
the energy exerted in such unit of light ; but it may be 
long before the relation is exactly determined. 

A provisional unit, then, means one which is assumed 
and physically defined in a safe and reproducible manner, 
in order that particular quantities may be compared inUr 
96 more accurately than they can yet be referred to the 
primary units. In reality the great majority of our 
measurements are expressed in terms of such provisionally 
independent units, and even the unit of mass, as we have 
seen, ought to be considered as provisional. 

The unit of heat ought to be simply the unit of energy, 
already described. But a weight can be measured to the 

' rhilosophical Trantactiom (1859), vol. cxlix. p. 884, &c. 

one- millionth part, and temperature to less than the 
thousandth part of a degree Fahrenheit, and to less there- 
fore than the five-hundred thousandth part of the absolute 
temperature, whereas the mechanical equivalent of heat is 
probably not known to the thousandth part. Hence the 
need of a provisional unit of heat, which is often taken as 
that requisite to raise one gram of water through one degree 
Centigrade, that is from 0° to 1°. This quantity of heat is 
capable of approximate expression in terms of time, space, 
and mass ; for by the natural constant, determined hy Dr! 
Joule, and called the mechanical equivalent of heat, we 
know that the assumed unit of heat is equal to the energy 
0^ 423*55 gram-metres, or that energy which will raise 
the mass of 423*55 grams through one metre against 9-8... 
absolute units of force. Heat, may also be expressed in 
terms of the quantity of ice at 0° Cent., which it is capable 
of converting into water under inappreciable pressure. 

Theory of Dimeimons, 

In order to understand the relations between the quan- 
tities dealt with in physical science, it is necessary to pay 
attention to the Theory of Dimensions, first clearly stated 
by Joseph Fourier,^ but in later years developed by several 
physiciste. This theory investigates the manner in which 
each derived unit depends upon or involves one or more of 
the fundamental units. The number of units in a rectan- 
gular area is found by multiplying together the numbers 
of units in the sides ; thus the unit of length enters twice 
into the unit of area, which is therefore said to have two 
dimensions with respect to length. Denoting length by L, 
we may say that the dimensions of area are Z x Z or 
Z«. It is obvious in the same way that the dimensions of 
volume or bulk will be L\ 

The number of units of mass in a body is found by nml- 
tiplying the number of units of volume, by those of density. 
Hence mass is of three dimensions as regards length, 
and one as regards density. Calling density D, the dimen- 
sions of mass are DD, As already explained, however, 
It is UBual to substitute an arbitrary provisional unit of 

• TKiom Aiudytiqiu cU la Chaleur, Paris; 1822, §§ 157- i6t. 




f j 


mass, symbolised by M ; according to the view here taken 
we may say that the dimensions of M are Z*i). 

Introducing time, denoted by T, it is easy to see that 

the dimensions of velocity will be — or LT-^, because 

the number of units in the velocity of a body is found 
by dividing the units of length passed over by the units 
of time occupied in passing. The acceleration of a body 
is measured by the increase of velocity in relation to 
the time, that is, we must divide the units of velocity 
gained by the units of time occupied in gaining it ; hence 
its dimensions will be LT'^. Momentum is the product 
of mass and velocity, so that its dimensions are MLT~\ 
The effect of a force is measured by the acceleration 
produced in a unit of mass in a unit of time ; hence the 
dimensions of force are MLT'^. Work done is pro- 
portional to the force acting and to the space through 
which it acts ; so that it has the dimensions of force with 
that of length added, giving ML^T-\ 

It should be particularly noticed that angular mag- 
nitude has no dimensions at all, being measured by the 
ratio of the arc to the radius (p. 305). Thus we have the 
dimensions LL'^ or L^. This^rees with the statement 
previously made, that no arbitrary unit of angular mag- 
nitude is needed. Similarly, all pure numbers expressing 
ratios only, such as sines and other trigonometrical func- 
tions, logarithms, exponents, &c., are devoid of dimensions. 
They are absolute numbers necessarily expressed in terms 
of unity itself, and are quite unaffected by the selection of 
the arbitrary physical units. Angular magnitude, however, 
enters into other quantities, such as angular velocity, which 

has the dimensions —. or T-^, the units of angle being 

divided by the units of time occupied. The dimensions of 
angular acceleration are denoted by T"*. 

The quantities treated in the theories o,' heat and 
electricity are numerous and complicated as regards 
*hfclr dimensions. Thermal capacity has the dimensions 
ML~^, thermal conductivity, ML-^T~\ In Magnetism 
the dimensions of the strength of pole are AOL^T-\ 
the oip^ensions of ne^d-intensitv are M ^L~^T~\ and the 


intensity of magnetisation has the same dimensions. In the 
science of electricity physicists have to deal with numerous 
kinds of quantity, and their dimensions are different too in 
the electro-static and the electro-magnetic systems. Thus 
electro - motive force has the dimensions M^L^T \ in 
the former, and M^DT^ in the latter system. Capa- 
city simply depends upon length in electro- statics, but 
upon Zf~*r» in electro-magnetics. It is worthy of par- 
ticular notice that electrical quantities have simple dimen- 
sions when expressed in terms of density instead of mass. 
The instances now given are sufficient to show the diffi- 
culty of conceiving and following out the relations of the 
quantities treated in physical science without a systematic 
method of calculating and exhibiting their dimensions. It 
is only in quite recent years that clear ideas about these 
quantities have been attained. Half a century ago pro- 
bably no one but Fourier could have explained what he 
meant by temperature or capacity for heat. The notion 
of measuring electricity had hardly been entertained. 

Besides affording us a clear view of the complex relations 
of physical quantities, this theory is specially useful in 
two ways. Firstly, it affords a test of the correctness of 
mathematical reasoning. According to the Principle of 
Homogeneity, all the quantities added together, and equated 
in any equation, must have the same dimensions. Hence 
if, on estimating the dimensions of the terms in any equa- 
tion, they be not homogeneous, some blunder must have 
been committed. It is impossible to add a force to a velo- 
city, or a mass to a momentum Even if the numerical 
values of the two members of a non-homogeneous equation 
were equal, this would be accidental, and any alteration in 
the physical units would produce inequality and disclose 
the fakity of the law expressed in the equation. 

Secondly, the theory of units enables us readily and 
infallibly to deduce the change in the numerical expression 
of any physical quantity, produced by a change in the 
fundamental units. It is of course obvious that in order 
to represent the same absolute quantity, a number must 
vary inversely as the magnitude of the units which are 
numbered. The yard expressed in feet is 3 ; taking th<i 
inch as the unit instead of the foot it becomes 36. Every 
quantity into which the dimension length enters pcaitivoly 




M • 




must be altered in like manner. Changing the unit from 
the foot to the inch, numerical expressions of volume must 
be multiplied by 12 x 12 x 12. When a dimension enters 
negatively the opposite rule will hold. If for the minute 
we substitute the second as unit of time, then we must 
divide all numbers expressing angular velocities by 60, 
and numbers expressing angular acceleration by 60 x 60. 
The rule is that a numerical expression varies inversely as 
the magnitude of the unit as regards each whole dimension 
entering positively, and it varies directly as the magnitude 
of the unit for each whole dimension entering negatively. 
In the case of fractional exponents, the proper root of the 
ratio of change has to be taken. 

The study of this subject may be continued in Professor 
J. D. Everett's " Illustrations of the Centimetre-gramme- 
second System of Units," published by Taylor and Francis, 
1875 ; in Professor Maxwell's " Theory of Heat ; " or Pro- 
fessor Fleeming Jenkin's " Text Book of Electricity." 

Natural Constaiits. 

Having acquired accurate measuring instruments, and 
decided upon the units in which the results shall be ex- 
pressed, there remains the question, What use shall be 
made of our powers of measurement ? Our principal 
object must be to discover general quantitative laws of 
nature ; but a very large amount of preliminary labour is 
employed in the accurate determination of the dimensions 
of existing objects, and the numerical relations between 
diverse forces and phenomena. Step by step every part 
of the material universe is surveyed and brought into 
known relations with other parts. Each manifestation of 
energy is correlated with each other kind of manifestation. 
Professor Tyndall has described the care with which such 
operations are conducted.^ 

" Those who are unacquainted with the details of 
scientific investigation, have no idea of the amount of 
labour expended on the determination of those numbers 
on which important calculations or inferences depend. 
They have no idea of the patience shown by a Berzelius 
in determining atomic weights ; by a Regnault in deter- 

* Tyndall's Sound, irt ed. p. 26. 


mining coefficients of expansion ; or by a Joule in deter- 
mining the mechanical equivalent of heat. There is a 
morality brought to bear upon such matters which, in 
point of severity, is probably without a parallel in any other 
domain of intellectual action." 

Eveiy new natural constant which is recorded brings 
many fresh inferences within our power. For if n be the 
number of such constants known, then J (v? — n) is the 
number of ratios which are within our powers of calcula- 
tion, and this increases with the square of n. We thus 
gradually piece together a map of nature, in which the 
lines of inference from one phenomenon to another rapidly 
grow in complexity, and the powers of scientific prediction 
are correspondingly augmented. 

Babbage 1 proposed the formation of a collection of the 
constant numbers of nature, a work which has at last 
been taken in hand by the Smithsonian Institution.^ It 
is true that a complete collection of such numbers would 
be almost co-extensive with scientific literature, since 
almost all the numbers occurring in works on chemistry, 
mineralogy, physics, astronomy, &c., would have to be 
included. Still a handy volume giving all the more 
important numbers and their logarithms, referred when 
requisite to the different units in common use, would be 
very useful. A small collection of constant numbers will 
be found at the end of Babbage's, Button's, and many 
other tables of logarithms, and a somewhat larger collec- 
tion is given in Templeton's Millvjright and Engineer'i 
Potket Companion, 

Our present object will be to classify these constant 
numbers roughly, according to their comparative generality 
and importance, under the following heads : — 
(i) Mathematical constants. 

(2) Physical constants. 

(3) Astronomical constants. 

i4) Terrestrial numbers. 
5) Organic numbers. 
(6) Social numbers. 

1 f "!:''?'* Aeeociation, Cambridge, 1833. Report, pp. 484—490. 
bmxlhtonxan Miscellaneous Collections, vol. xii., the Constants of 
mature, part. 1. Specific gravities compiled by F. W. Clarke. 8vo. 
washmgton, 1873. 


\ I 


1 1 






If : 





in ' 

! , 













Mathematical Constants, 


At the head of the list of natural constants must come 
those which express the necessary relations of numbers to 
each other. The ordinary Multiplication Table is the 
most familiar and the most important of such series of 
constants, and is, theoretically speaking, infinite in extent. 
Next we must place the Arithmetical Triangle, the sig- 
nificance of which has already been pointed out (p. 182). 
Tables of logarithms also contain vast series of natural 
constants, arising out of the relations of pure numbers. 
At the base of all logarithmic theory is the mysterious 
natural constant commonly denoted by e, or e, being 

equal to the infinite series i + - + ' + — — I \- 

^ 1 ^ 1.2^ 1.2.3 ^^ ' 

and thus consisting of the sum of the ratios between the 
numbers of permutations and combinations of o, i, 2, 3, 
4, &c. things. Tables of prime numbers and of the factors 
of composite numbers must not be forgotten. 

Another vast and in fact infinite series of numerical 
constants contains those connected with the measure- 
ment of angles, and embodied in trigonometrical tables, 
whether as natural or loganthmic sines, cosines, and 
tangents. It should never be forgotten that though 
these numbers find their chief employment in connection 
with trigonometry, or the measurement of the sides of a 
right-angled triangle, yet the numbers themselves arise 
out of numerical relations bearing no special relation to 
space. Foremost among trigonometrical constants is the 
well known number ir, usually employed as expressing 
the ratio of the circumference and the diameter of a 
circle ; from tr follows the value of the arcual or natural 
unit of angular value as expressed in ordinary degrees 
(p. 306). 

Among other mathematical constants not uncommonly 
used may be mentioned tables of factorials (p. 179), tables 
of Bemouilli's numbers, tables of the error function,^ 
which latter are indispensable not only in the theory of 
probability but also in several other branches of science. 

1 J. W. L. Qlaisher, PhiUnophical Magaavu, 4th Series, yoL xlii 
p. 421. 


It should be clearly underatood that the mathematical 
constants and tables of reference already in our possession, 
although very extensive, are only an infinitely small part 
of what might be formed. With the progress of science 
the tabulation of new functions will be continually 
demanded, and it is worthy of consideration whether 
public money should not be available to reward the 
severe, long continued, and generally thankless labour 
which must be gone through in calculating tables. Such 
labours are a benefit to the whole human race as long as 
it shall exist, though there are few who can appreciate 
the extent of this benefit. A most intere«sting and excel- 
lent description of many mathematical tables will be 
found in De Morgan's article on Tables^ in the English 
Cyclopmdia, Division of Arts and Sciences, vol. vii. p. 976. 
An almost exliaustive critical catalogue of extant tables is 
being published by a Committee of the British Association, 
two portions, drawn up chiefly by Mr. J. W. L Glaisher 
and Professor Cayley, having appeared in the Reports of 
the Association for 1873 and 1875. 

Physical Constants. 

The second class of constants contains those which 
refer to the actual constitution of matter. For the most 
part they depend upon the peculiarities of the chemical 
substance in question, but we may begin with those 
which are of the most general character. In a first sub- 
class we may place the velocity of light or heat undula- 
tions, the numbers expressing the relation between the 
lengths of the undulations, and the rapidity of the 
undulations, these numbers depending only on the pro- 
perties of the ethereal medium, and being probably the 
same in all parts of the universe. The theory of heat 
gives rise to several numbers of the highest importance, 
especially Joule's mechanical equivalent of heat, the 
absolute zero of temperature, the mean temperature of 
empty space, &c. 

Taking into account the diverse properties of the 
elements we must have tables of the atomic weights, 
the specific heats, the specific gravities, the refractive 
powers, not only of the elements, but their almost 




infinitely numerous compounds. The properties of hardness, 
elasticity, viscosity, expansion by heat, conducting powers 
for heat and electricity, must also be determined in 
immense detail. There are, however, certain of these 
numbers which stand out prominently because they serve 
as intermediate units or terms of comparison. Such are, 
for instance, the absolute coefficients of expansion of air, 
water and mercury, the temperature of the maximum 
density of water, the latent heats of water and steam, 
the boiling-point of water under standard pressure, the 
melting and boiling-points of mercury, and so forth. 

Astronomical Constants. 

The third great class consists of numbers possessing far 
less generality because they refer not to the properties of 
matter, but to the special forms and distances in which 
matter has been disposed in the part of the universe open 
to our examination. We have, first of all, to define the 
magnitude and form of the earth, its mean density, the 
constant of aberration of light expressing the relation 
between the earth's mean /velocity in space and the 
velocity of light. From the earth, as our observatory, we 
then proceed to lay down the mean distances of the sun, 
and of the planets from the same centre ; all the elements 
of the planetary orbits, the magnitudes, densities, masses, 
periods of axial rotation of the several planets are by 
degrees determined with growing accuracy. The same 
labours must be gone through for the satellites. Cata- 
logues of comets with the elements of their orbits, as far 
as ascertainable, must not be omitted. 

From the earth's orbit as a new base of observations, 
we next proceed to survey the heavens and lay down the 
apparent positions, magnitudes, motions, distances, periods 
of variation, &c. of the stars. All catalogues of stars from 
those of Hipparchus and Tycho, are fuU of numbers ex- 
pressing rudely the conformation of the visible universe. 
But there is obviously no limit to the labours of astrono- 
mers ; not only are millions of distant stars awaiting their 
first measurements, but those already registered require 
endless scrutiny as regards their movements in the three 
dimensions of space, their periods of revolution, their 

xitJ units and standards of MEASUREMENT. 338 

changes of brilliance and colour. It is obvious that 
though astronomical numbers are conventionally called 
constant, they are probably in all cases subject to more 
or less rapid variation. 

Terrestrial Numbers. 

Our knowledge of the globe we inhabit involves many 
numerical determinations, which have little or no con- 
nection with astronomical theory. The extreme heights 
of the principal mountains, the mean elevations of 
continents, the mean or extreme depths of the oceans, 
the specific gravities of rocks, the temperature of mines, 
the host of numbers expressing the meteorological or 
magnetic conditions of every part of the surface, must 
fall into this class. Many such numbers are not to be 
called constant, being subject to periodic or secular 
changes, but they are hardly more variable in fact than 
some which in astronomical science are set down as 
constant. In many cases quantities which seem most 
variable may go through rhythmical changes resulting 
in a nearly uniform average, and it is only in the long 
progress of physical investigation that we can hope to 
discriminate successfully between those elemental num- 
bers which are fixed and those which vary. In the latter 
case the law of variation becomes the constant relation 
which is the object of our search. 

Organic Numbers. 

The forms and properties of brute nature having be«»Ji 
sufficiently defined by the previous classes of numbers, 
the organic world, both vegetable and animal, remains 
outstanding, and offers a higher series of phenomena for 
our investigation. All exact knowledge relating to the 
forms and sizes of living things, their numbsrs, the 
quantities of various compounds which they consume, 
contain, or excrete, their muscular or nervous energy, &c. 
must be placed apart in a class by themselves. All such 
numbers are doubtless more or less subject to variation, 
and but in a minor degree capable of exact determination. 
Man, so far as he is an animal, and as regards his physic/al 
form, must also be treated in this class. 




Social Numbers, 

Little allusion need be made in this work to the fact 
that man in his economic, sanitary, intellectual, aesthetic, 
or moral relations may become the subject of sciences, 
the highest and most usefiil of all sciences. Every one 
who is engaged in statistical inquiry must acknowledge 
the possibflity of natural laws governing such statistical 
facts. Hence we must allot a distinct place to numerical 
information relating to the numbers, ages, physical and 
sanitary condition, mortality, &c., of difiFerent peoples, in 
short, to vital statistics. Economic statistics, compre^ 
bending the quantities of commodities produced, existing, 
exchanged and consumed, constitute another extensive 
body of science. In the progress of time exact investi- 
gation may possibly subdue regions of phenomena which 
at present defy all scientific treatment That scientific 
method can ever exhaust the phenomena of the human 
mind is incredible. 

:.i h 





I I 



. ,' . ' 'J 

In the two preceding chapters we have been eng^ed 
in considering how a phenomenon may be accurately 
measured and expressed. So delicate and complex an 
operation is a measurement which pretends to any con 
siderable degree of exactness, that no small part of the 
skill and patience of physicists is usually spent upon this 
work. Much of this diflSculty arises from the fact that 
it is scarcely ever possible to measure a single effect at a 
time. The ultimate object must be to discover the 
mathematical equation or law connecting a quantitative 
cause with its quantitative effect ; this purpose usually 
involves, as we shall see, the varying of one condition at 
a time, the other conditions being maintained constant 
The labours of the experimentalist would be compara- 
tively light if he could carry out this rule of varying one 
circumstance at a time. He would then obtain a series of 
corresponding values of the variable quantities concerned, 
from which he might' by proper hypothetical treatment 
obtain the required law of connection. Bi^ in reality it 
is seldom possible to carry out this direction except in an 
approximate manner. Before then we proceed to the 
consideration of the actual process of quantitative induc- 
tion, it is necessary to review the several devices by 
which a complicated series of effects can be disentangled. 
Every phenomenon measured will usually be the sum, 
difference, or it may be the product or quotient, of 
two or more different effects, and these must be in some 

















way analysed and separately measured before we possess 
the materials for inductive treatment. 

lUmtrations of the Complication of Effects, 

It is easy to bring forward a multitude of instances to 
show that a phenomenon is seldom to be observed simple 
and alone. A more or less elaborate process of analysis 
is almost always necessary. Thus if an experimentalist 
wishes to observe and measure the expansion of a liquid 
by heat, he places it in a thermometer tube and registers 
the rise of the column of liquid in the narrow tube. But 
he cannot heat the liquid without also heating the glass, 
80 that the change observed is really the difference between 
the expansions of the liquid and the glass. More minute 
investigation will show the necessity perhaps of allowing 
for further minute effects, namely the compression of the 
liquid and the expansion of the bulb due to the increased 
pressure of the column as it becomes lengthened. 

In a great many cases an observed effect will be 
apparently at least the simple sum q£ two separate and 
independent effects. The hea^ evolved in the combustion 
of oil is partly due to the carbon and partly to the 
hydrogen. A measurement of the heat yielded by the two 
jointly, cannot inform us how much proceeds from the 
one and how much from the other. If by somcseparate 
determination we can ascertain how much the hydrogen 
yields, then by mere subtraction we learn what is due 
to the carbon; and vice vtrsd. The heat conveyed by a 
liquid, may be partly conveyed by true conduction, partly 
by convection. The light dispersed in the interior of a 
liquid consists both of what is reflected by floating 
particles and what is due to true fluorescence;^ and we 
must find some mode of determining one portion before 
we can learn the other. The apparent motion of the spots 
on the sun, ia the algebraic sum of the sun's axial 
rotation, and^of the proper motion of the spots upon the 
sun's surface; hence the difficulty of ascertaining by 
direct observations the period of the sun's rotation. 

We cannot obtain the weight of a portion of liquid 

• Stokes, Ph%l4>topliieal Trantacti&yis (1852), vo*. cxiii. p. 529. 


in a chemical balance without weighing it with the 
containing vessel. Hence to have the real weight of 
the liquid operated upon in an experiment, we must 
make a separate weighing of the vessel, with or without 
the adhering film of liquid according to circumstances. 
This is likewise the mode in which a cart and its load 
are weighed together, the tare of the cart previously 
ascertained being deducted. The variation in the height 
of -the barometer is a joint effect, partly due to the real 
variation of the atmospheric pressure, partly to the expan- 
sion of the mercurial column by heat. The effects may 
be discriminated, if, instead of one barometer tube we have 
two tubes containing mercury placed closely side by side, 
so as to have the same temperature. If one of them be 
closed at the bottom so as to be unaffected by the atmo- 
spheric pressure, it will show the changes due to tempera- 
ture only, and, by subtracting these changes from those 
shown in the other tube, employed as a barometer, we 
get the real oscillations of atmospheric pressure. But 
this correction, as it is called, of the barometric reading, 
is better effected -by calculation from the readings of 
an ordinary thermometer. 

In other cases a quantitative effect will be the difference 
of two causes acting in opposite directions. Sir John 
Herschel invented an instrument like a large thermometer, 
which h« called the Actinometer,^ and Pouillet constructed 
a somewhat similar instrument called the Pyrheliometer, 
for ascertaining the heating power of the sun's rays. In 
both instruments the heat of tlie sun was absorbed by a 
reservoir containing water, and the rise of temperature 
of the water was exactly observed, either by its own 
expansion, or by the r^adings of a delicate thermometer 
immersed in it But in exposing the actinometer to the 
sun, we do not obtain the full effect of the heat absorbed, 
because the receiving surface is at the same time radiating 
heat into empty space. The observed increment of tem- 
perature is in short the difference between what is received 
from the sun and lost by radiation. The latter quantity is 
capable of ready determination ; we have only to shade the 
instrument from the direct rays of the sun, leaving it 

• Admiralty Manual of Scientific Enquiry ^ 2nd ed. p. 299. 




exposed to the sky, and we can observe how much it cools 
in a certain time. The total effect of the sun's rays will 
obviously be the apparent effect pliis the cooling effect in 
an equal time. By alternate exposure in sun and shade 
during equal intervals the desired result may be obtained 
with considerable accuracy.^ 

Two quantitative effects were beautifully distinguished 
in an experiment of John Canton, devised in 1761 for the 
purpose of demonstrating the compressibility of water. 
He constructed a thermometer with a large bulb full of 
water and a short capillary tube, the part of which above 
the water was freed from air. Under these circumstances 
the water was relieved from the pressure of the atmo- 
sphere, but the glass bulb in bearing that pressure was 
somewhat contracted. He next placed the instrument 
under the receiver of an air-pump, and on exhausting the 
air, the water sank in the tube. Having thus obtained a 
measure of the effect of atmospheric pressure on the bulb, 
he opened the top of the thermometer tube and admitted 
the air. The level of the water now sank still more, partly 
from the pressure on the buU) being now compensated, and 
partly from the compression of the water by the atmo- 
spheric pressure. It is obvious that the amount of the 
latter effect was approximately the difference of the two 
observed depressions. 

Not uncommonly the actual phenomenon which we wish 
to measure is considerably less than various disturbing 
effects which enter into the question. Thus the compres- 
sibility of mercury is considerably less than the expansion 
of the vessels in which it is measured under pressure, so 
that the attention of the experimentalist has chiefly to be 
concenti-ated on the change of magnitude of the vessels. 
Many astronomical phenomena, such as the parallax or the 
proper motions of the fixed stars, are far less than the 
errors caused by instrumental imperfections, or motions 
arising from precession, nutation, and aberration. We 
need not be surprised that astronomers have from time to 
time mistaken one phenomenon for another, as when Flam- 
steed ifjiagined that he had discovered the parallax of the 
Pole star.^ 

* Pottillet, Taylor^s Seienti/k Memoirs^ vol. iy. p. 45. 
' BaUy'a AocohtU of the Hev. John Flartuteed, p. 58. 



Methods of Eliminating Error, 

In any particular experiment it is the object of the ex- 
perimentalist to measure a single effect only, and he 
endeavours to obtain that effect free from interfering^ 
effects. If this cannot bo, as it seldom or never can 
really be, he makes the effect as considerable as possible 
compared with the other effects, which he reduces to a 
minimum, and treats as noxious errors. Those quantities, 
which are called errors in one case, may really be most 
important and interesting phenomena in another investiga- 
tion. When we speak of eliminating error we really 
mean disentangling the complicated phenomena of nature 
The physicist rightly wishes to treat one thing at a time, 
but as this object can seldom be rigorously carried into 
practice, he has to seek some mode of counteracting the 
irrelevant and interfering causes. ^ 

The general principle is that a single observation can 
render known only a single quantity. Hence, if several 
different quantitative effects are known to enter into any 
investigation, we must have at least as many distinct ob- 
servations as there are quantities to be determined. Every 
complete experiment wQl therefore consist in ^reneral of 
several operations. Guided if possible by previous know- 
ledge of the causes in action, we must arrange the deter- 
minations, so that by a simple mathematical process we 
may distinguish the separate quantities. There appear to 
be five principal methods by which we may accomplish 
this object ; these methods are specified below and illus- 
I trated in the succeeding sections. 

(i) The Method of Avoidance. The physicist may seek 
for some special mode of experiment or opportunity of obser- 
vation, in which the error is non-existent or inappreciable. 

(2) The Differential Method. He may find opportunities 
of observation when all interfering phenomena remain con 
stant, and only the subject of observation is at one time 
present and another time absent; the difference between 
two observations then gives its amount 

(3) The Mahod of Correction. He may endeavour to 
estimate the amount of the interfering effect by the best 
available mode, and then make a corresponding correction 
ui the lesults of observation. 

2 2 




' I 

(4) The Method of Compensation. He may invent some 
mode of neutralising the interfering cause by balancing 
against it an exactly equal and opposite cause of unknown 

(5) The Method of Reversal. He may so conduct the 
experiment that the interfering cause may act in opposite 
directions, in alternate observations, the mean result being 
free from interference. 


I . Method of Avoidance of Error, 

Astronomers seek opportunities of observation when 
errors wOl be as small as possible. In spite of elaborate 
observations and long-continued theoretical investigation, 
it is not practicable to assign any satisfactory law to the 
refractive power of the atmosphere. Although the appa- 
rent change of place of a heavenly body produced by 
refraction may be more or less accurately calculated yet 
the error depends upon the temperature and pressure of 
the atmosphere, and, when a ray is highly inclined to the 
pei*pendicular, the uncertaifHy in the refraction becomes 
very considerable. Hence astronomers always make their 
observations, if possible, when the object is at the highest 
point of its daily course, i.e. on the meridian. In some 
kinds of investigation, as, for instance, in the determination 
of the latitude of an observatory, the astronomer is at 
liberty to select one or more stars out of the countless 
number visible. There is an evident advantage in such a 
case, in selecting a star which passes close to the zenith, 
so that it may he observed almost entirely free from atmo- 
spheric refraction, as was done by Hooke. 

Astronomers endeavour to render their clocks as accurate 
as possible, by removing the source of variation. The 
pendulum is perfectly isochronous so long as its length 
remains invariable, and the vibrations are exactly of equal 
length. They render it nearly invariable in length, that 
is in the distance between the centres of suspension and 
oscillation, by a compensatory arrangement for the change 
of temperature. But as tliis compensation may not be 
perfectly accomplished, some astronomers place their chief 
controlling clock in a cellar, or other apartment, where 
the changes of temperature may be as slight as possible. 

«.n-. . 


At the Paris Observatory a clock has been placed in the 
caves beneath the building, where there is no appreciable 
difference between the sunimer and winter temperature. 

To avoid the effect of unequal oscillations Huyghens 
made his beautiful investigations, which resulted in the 
discovery that a pendulum, of which the centre of oscilla- 
tion moved upon a cycloidal path, would be perfectly 
isochronous, whatever the variation in the length of oscilla- 
tions. But though a pendulum may be easily rendered in 
some degree cycloidal by the use of a steel suspension 
spring, it is found that the mechanical arrangements re- 
quisite to produce a truly cycloidal motion introduce moi-e 
error than they remove. Hence astronomers seek to 
reduce the error to the smallest amount by maintaining 
their clock pendulums in uniform movement; in fact, 
while a clock is in good order and has the same weights, 
there need be little change in the length of oscillation. 
When a pendulum cannot be made to swing uniformly, as 
in experiments upon the force of gravity, it becomes re- 
quisite to resort to the third method, and a correction is 
introduced, calculated on theoretical grounds from the 
amount of the observed change in the length of vibration. 

It has been mentioned that the apparent expansion of a 
liquid by heat, when contained in a thermometer tube or 
other vessel, is the difference between the real expansion 
of the liquid and that of the containing vessel The 
effects can be accurately distinguished provided that we 
can learn the real expansion by heat of any one convenient 
liquid ; for by observing the apparent expansion of tlio 
same liquid in any required vessel we can by difference 
learn the amount of expansion of the vessel due to any 
given change of temperature. When we once know the 
change of dimensions of the vessel, we can of course deter- 
mine the absolute expansion of any other liquid tested in 
it Thus it became an all-important object in scientific 
research to measure with accuracy the absolute dilatation 
by heat of some one liquid, and mercury owing to several 
circumstances was by far the most suitable. Dulong and 
Petit devised a beautiful mode of effecting this by simply 
avoiding altogether the effect of the change of size of the 
vessel Two upright tubes full of mercury were connected 
by a fine tube at the bottom, and were maintained at two 



\, f I 


il i, 




different temperatures. As mercury was free to flow from 
one tube to the other by the connecting tube, the two 
columns necessarily exerted equal pressures by the princi- 
ples of hydrostatics. Hence it was only necessary to mea- 
sure very accurately by a cathetometer the difference of 
level of the surfaces of the two columns of mercury, to 
learn the difference of length of columns of equal hydro- 
static pressure, which at once gives the difference of den- 
sity of the mercury, and the dilatation by heat. The 
changes of dimension in the containing tubes became a 
matter of entire indifference, and the length of a colunni 
of mercury at different temperatures was measured as 
easily as if it had formed a solid bar. The experiment was 
carried out by Regnault with many improvements of detail, 
and the absolute dilatation of mercury, at temperatures 
between o° Cent and 350°, was determined almost as 
accurately as was needful^ 

The presence of a large and unceitain amount of eiTor 
may render a method of experiment valueless. Foucault 
devised a beautiful experilnent with the pendulum for 
demonstrating popularly the rotation of the earth, but it 
could be of no use for measuring the rotation exactly. It 
is impossible to make the pendulum swing in a perfect 
plane, and the slightest lateral motion gives it an elliptic 
path with a progressive motion of the axis of the ellipse, 
wliich disguises and often entirely overpowers that due to 
the rotation of the earth.* 

Faraday's laborious experiments on the relation of gravity 
and electricity were much obstructed by the fact that it is 
impossible to move a large weight of metal without gener- 
ating currents of electricity, either by friction or induction. 
To distinguish the electricity, if any, directly due to the 
action of gravity from the greater quantities indirectly pi-o- 
duced was a problem of excessive difficulty. Baily in his 
experiments on the density of the earth was aware of the 
Bxistence of inexplicable disturbances which have since 
been referred with much probability to the action of 
electricity.* The skill and ingenuity of the experimentalist 

' Jamin, Court de Physique, vol. ii. pp. 15 — 28. 

• Fhilosophieal Magazine, 1851, 4th Series, vol. ii. pattim. 

* Heam, rMlotqphiccU 7VanM0<um«, 1847, vol. cxxxni. pp. 217 


are often exhausted in trying to devise a form of apparatus 
in which such causes of error shall be reduced to a 

In some rudimentary experiments we wish merely to 
establish the existence of a quantitative effect without 
precisely measuring its amount ; if there exist causes of 
error of which we can neither render the amount known 
or inappreciable, the best way is to make them all 
negative so that the quantitative effects will be less than 
the truth rather than gieater. Grove, for instance, in 
proving that the magnetisation or demagnetisation of a 
piece of iron raises its temperature, took care to maintain 
the electro-magnet by which the iron was magnetised at 
a lower temperature than the iron, so that it would cool 
rather than warm the iron by radiation or conduction.^ 

Humfoi-d's celebrated experiment to prove that heat was' 
generated out of mechanical force in the boring of a 
cannon was subject to the difficulty that heat might be 
brought to the cannon by conduction from neighbouring 
bodies. It was an ingenious device of Davy to produce 
friction by a piece of clock-work resting upon a block 
of ice in an exhausted receiver ; as the machine rose in 
temperature above 32°, it was certain that no heat was 
received by conduction from the support.^ In many 
other experiments ice may be employed to prevent the 
access of heat by conduction, and this device, first put in 
practice by Murray,* is beautifully employed in Bunsen's 

To observe the true temperature of the air, though 
apparently so easy, is really a very difficult matter, because 
the thermometer is sure to be aft'ected either by the sun's 
rays, the radiation from neighbouring objects, or the escape 
of heat into space. These sources of enor are too fluctu- 
ating to allow of correction, so that the only accurate mode 
of procedure is that devised by Dr. Joule, of surrounding 
the thermometer with a copper cylinder ingeniously 

* The Corrdation of Physical Forces^ ^rd ed. p. 159. 

* Collected Works of Sir M, Davy, vol. ii. pp. 12—14. Elements of 
Chemical Philosophy, p. 94. 

^ Nicholson's Journal, vol. i. p. 241 ; quoted in Treatiu on Ileat 
Useful Knowledge Society, p. 24. 




adjusted to the temperature of the air, as described by 
him, so that the effect of radiation shall be nullified.^ 

When the avoidance of error is not practicable, it will 
yet be desirable to reduce the absolute amount of the 
interfering error as much as possible before employing the 
succeeding methods to correct the result. As a general 
rule we can determine a quantity with less inaccuracy as 
it is smaller, so that if the eri-or itself be small the error in 
determining that error will be of a still lower order of 
magnitude. But in some cases the absolute amount of an 
error is of no consequence, as in the index error of a 
divided circle, or the difference between a chronometer and 
astronomical time. Even the rate at which a clock gains 
or loses is a matter of little importance provided it remain 
constant, so that a sure calculation of its amount can be 


2. Differential Method, 

When we cannot avoid the existence of error, we can 
often resort with success to the second mode by measuring 
phenomena under such circumstances that the eiTor shall 
remain very nearly the same in all the observations, and 
neutralise itself as regards the purposes in view. This 
mode is available whenever we want a difference between 
quantities and not the absolute quantity of either. The 
determination of the parallax of the fixed stars is exceed- 
ingly difficult, because the amount of parallax is far less 
than most of the corrections for atmospheric refraction, 
nutation, aberration, precession, insti-umental irregularities, 
&c., and can with difficulty be detected among these pheno- 
mena of various magnitude. But, as Galileo long ago 
suggested, all such difficulties would be avoided by the 
differential observation of stars, which, though apparently 
close together, are really fai separated on the line of sight. 
Two such stars in close apparent proximity will be sub- 
ject to almost exactly equal errors, so that all we 
need do is to observe the apparent change of place of 
the nearer star as referred to the more distant one. 

J ^1*^^^ ^IS'^)''^^^\ T^^"^ ^f ^^f P 228. Proceedings of the 
Manchester Phxlos(n^k%9al Society, Nov. 26. 1867, vol. vii. p' 35. 


A good telescope furnished with an accurate micrometer 
is alone needed for the application of the method. 
Huyghens appears to have been the first observer who 
actually tried to employ the method practically, but 
it was not until 1835 that the improvement of telescopes 
and micrometers enabled Struve to detect in this way 
the parallax of the star a Lyrae. It is one of the many 
advantages of the observation of transits of Venus for the 
determination of the solar parallax that the refraction of 
the atmosphere affects in an exactly equal degree the planet 
and the portion of the sun's face over which it is passing. 
Thus the observations are strictly of a differential nature. 

By the process of substitutive weighing it is possible 
to ascertain the equality or inequality of two weights 
with almost perfect freedom from error. If two weights 
A and B be placed in the scales of the best balance 
vre cannot be sure that the equilibrium of the beam 
indicates exact equality, because the arms of the beam 
may be unequal or unbalanced. But if we take B out 
and put another weight C in, and equilibrium still 
exists, it is apparent that the same causes of erroneous 
weighing exist in both cases, supposing that the balance 
has not been disarranged ; B then must be exactly equal 
to C, since it has exactly the same effect under the same 
circumstances. In like manner it is a general rule that, 
if by any uniform mechanical process we get a copy of an 
object, it is unlikely that this copy will be precisely the 
same as the original in magnitude and form, but two copies 
will equally diverge from the original, and will therefore 
almost exactly resemble each other. 

Leslie's Differential Thermometer ^ was well adapted 
to the experiments for which it was invented. Having 
two equal bulbs any alteration in the temperature of the 
air will act equally by conduction on each and produce 
no change in the indications of the instrument. Only 
that radiant heat which is purposely thrown upon one 
of the bulbs will produce any effect. This thermometer 
in short carries out the principle of the differential method 
in a mechanical manner. 

' Leslie, Inquiry into tht Nature of Heat, p. la 





3. Method of Con^ectum, 

Whenever the result of an experiment is affected by an 
interfering cause to a calculable amount, it is sufficient to 
add or subtract this amount We are said to correct 
observations when we thus eliminate what is due to 
extraneous causes, although of course we are only sepa- 
rating the correct effects of sevend agents. The variation 
in the height of the barometer is partly due to the change 
of temperature, but since the coefficient of absolute 
dilatation of mercury has been exactly determined, as 
already described (p. 341), we have only to make cal- 
culations of a simple character, or, what is better still, 
tabulate a scries of such calculations for general use, and 
the correction for temperature can be made with all desired 
accuracy. The height of the mercury in the barometer is 
also affected by capillary attraction, which depresses it by 
a constant amount depending mainly on the diameter of 
the tube. The requisite corrections can be estimated with 
accuracy sufficient for most purposes, more especially as 
we can check the correctness of the reading of a barometer 
by comparison with a standard barometer, and introduce 
if need be an index error including both the error in the 
affixing of the scale and the effect due to capillarity. But 
in constructing the standard barometer itself we must take 
greater precautions; the capillaiy depression depends 
somewhat upon the quality of the glass, the absence of air, 
and the perfect cleanliness of the mercury, so that we 
cannot assign the exact amount of the effect Hence a 
standard barometer is constructed witli a wide tube, some- 
times even an inch in diameter, so that the capillary effect 
may be rendered almost zero.^ Gay-Lussac made baro- 
meters in the form of a uniform siphon tube, so that the 
capillary forces acting at the upper and lower surfaces 
should balance and destroy each other ; but the method 
fails in practice because the lower surface, being open to 
the air, becomes sullied and subject to a different force of 

In mechanical experiments fiiction is an interfering 
condition, and drains away a portion of the enei^ in- 

' Jevoiji, Watts' Z>K<umaiy 0/ CluiHutry, vol i pp. 5 1 3- 5 IS- 



tended to be operated upon in a definite manner. We 
should of course reduce the friction in the first place to the 
lowest possible amount, but as it cannot be altogether pre- 
vented, and is not calculable with certainty from any 
general laws, we must determine it separately for each 
apparatus by suitable experiments. Thus Smeaton, in 
his admirable but almost forgotten researches concerning 
water-wheels, eliminated friction in the most simple 
manner by determining by trial what weight, acting by a 
cord and roller upon his model water-wheel, would make 
it turn without water as i-apidly as the water made it turn. 
In short, he ascertained what weight concurring with the 
water would exactly compensate for the friction.^ In Dr. 
Joule's experiments to determine the mechanical equiva- 
lent of heat by the condensation of air, a considerable 
amount of heat was produced by friction of the condensing 
pump, and a small portion by stirring the water employed 
to absorb the heat This heat of friction was measured by 
simply repeating the experiment in an exactly similar 
manner except that no condensation was effected, and ob- 
serving the change of tenjperature then produced.^ 

We may describe as test experiments any in which we 
perform operations not intended to give the quantity of 
the principal phenomenon, but some quantity which would 
otherwise remain as an error in the result Thus in 
astronomical observations almost every instrumental error 
may be avoided by increasing the number of observations 
and distributing them in such a manner as to produce 
in the final mean as much error in one way as in the 
other. But there is one source of error, first discovei*ed 
by Maskelyne, which cannot be thus avoided, because it 
affects all observations in the same direction and to the 
same average amount, namely the Personal Error of the 
observer or the inclination to record the passage of a star 
across the wires of the telescope a little too soon or a 
little too late. This personal error was first carefully 
described in the Edinburgh Journal of Science, voL i. 
p. 1 78. The difference between the jud^ent of observers 
at the Greenwich Observatory usually varies from j^ to J 

I Philotophical TramactionSy vol. IL p. 100. 

* Philotophical Magazitu, 3rd Series, voL xxvl p. 372. 




i V 


of a second, and remains pretty constant for the same 
observers.* One practised observer in Sir George Airy's 
pendulum experiments recorded all his time observations 
half a second too early on the average as compared with 
the chief observer.* In some observers it has amounted to 
seven or eight- tenths of a second.' De Morgan appears to 
have entertained the opinion that this source of error was 
essentially incapable of elimination or correction.* But it 
seems clear, as I suggested without knowing what had 
been done,^ that this personal eiTor might be determined 
absolutely with any desirable degree of accuracy by test 
experiments, consisting in making an artificial star move 
at a considerable distance and recording by electricity the 
exact moment of its passage over the wire. This method 
has in fact been successfully employed in Leyden, Paris, 
and Neuchatel.* More recently, observers were trained 
for the Transit of Venus Expeditions by means of a 
mechanical model representing the motion of Venus over 
the sun, this model being placed at a little distance and 
viewed through a telescope, so that diil'erences in the 
judgments of different observers would become apparent. 
It seems likely that tests of this nature might be employed 
with advantage in other cases. 

Newton employed the pendulum for making experi- 
ments on the impact of balls. Two balls were hung in 
contact, and one of them, being drawn aside through a 
measured arc, was then allowed to strike the other, the 
arcs of vibration giving sufficient data for calculating the 
distribution of energy at the moment of impact The 
resistance of the air was an interfering cause which he 
estimated very simply by causing one of the balls to 
make several complete vibrations without impact and then 
marking the reduction in the lengths of the arcs^ a proper 
fraction of which I'eduction was added to each of the other 
arcs of vibration when impact took place.^ 

• Oreenwich Obtervatioiu for 1866, p. xlix. 

• Philosophical Transactions^ 1856, p. 309. 

• Fenny Cyclopadioj art. Transit, voL xxv. pp. 129, 13a 

• Ibid. art. OhservaJtion, p. 39a * Nature, vol. i. p. 85. 

• Nature, voL i. p. 337. See references to the Memoirs describing 
ftht method. 

' Frincipia, Book L Law III. Corollarj YL Scholiam. Motto't 
translatiou, vol. t p> 33. 



The exact definition of the standard of length is one 
of the most important, as it is one of the most difficult 
questions in physical science, and the different practice of 
different nations introduces needless confusion. Were 
all standards constructed so as to give the true length 
at a fixed uniform temperature, for instance the freezing- 
point, then any two standards could be compared without 
the interference of temperature by bringing them both 
to exactly the same fixed temperature. Unfortunately 
the French metre was defined by a bar of platinum at 
o*C, while our yard was defined by a bronze bar at 62°F. 
It is quite impossible, then, to make a comparison of the 
yard and metre without the introduction of a correction, 
either for the expansion of platinum or bronze, or both. 
Bars of metal differ too so much in their rates of ex- 
pansion according to their molecular condition that it is 
dangerous to infer from one bar to another. 

When we come to use instruments with great accuracy 
there are many minute sources of error which must be . 
guarded against. If a thermometer has been graduated 
when perpendicular, it will read somewhat differently 
when laid flat, as the pressure of a column of mercury 
is removed from the bulb. The reading may also be 
somewhat altered if it has recently been raised to a 
higher temperature than usual, if it be placed under a 
vacuous receiver, or if the tube be unequally heated as 
compared with the bulb. For these minute causes of 
error we may have to introduce troublesome corrections, 
unless we adopt the simple precaution of using the thermo- 
meter in circumstances of position, &c., exactly similar to 
those in which it was graduated. There is no end to 
the number of minute corrections which may ultimately 
be required. A large number of experiments on gases, 
standard weights and measures, &c., depend upon the 
height of the barometer ; but when experiments in dif- 
ferent parts of the world are compared together we ought 
as a further refinement to take into account the varying 
force of gravity, which even between London and Paris 
makes a difference of 'ooS inch of mercury. 

The measurement of quantities of heat is a matter of 
great difficulty, because there is no known substance 
impervious to heat, and the problem is therefore as 










difficult as to measure liquids in porous vessels. To 
determine the latent heat of steam we must condense a 
certain amount of the steam in a known weight of water, 
and then observe the rise of temperature of the water. 
But while we are carrying out the experiment, part of 
the heat will escape by radiation and conduction from 
the condensing vessel or calorimeter. We may indeed 
reduce the loss of heat by using vessels with double sides 
and bright surfaces, surrounded with swans-down wool or 
other non-conducting materials ; and we may also avoid 
raising the temperature of the water much above that of 
the surrounding air. Yet we cannot by any such means 
render the loss of heat inconsiderable. Rumford ingeni- 
ously proposed to reduce the loss to zero by commencing 
the experiment when the temperature of the calorimeter 
is as much below that of the air as it is at the end of the 
experiment above it Thus the vessel will first gain and 
then lose by radiation and conduction, and these opposite 
errors will approximately balance each other. But Reg- 
nault has shown that the loss and gain do not proceed by 
exactly the same laws, so that in very accurate inves- 
tigations Rumford's method is not sufficient There 
remains the method of correction which was beautifully 
carried out by Regnault in his determination of the latent 
heat of steauL He employed two calorimeters, made in 
exactly the same way and alternately used to condense a 
certain amount of steam, so that while one was measuring 
the latent heat, the other calorimeter was engaged in 
determining the corrections to be applied, whether on 
account of radiation and conduction from the vessel op 
on account of heat reaching the vessel by means of the 
connecting pipes.^ 

4. Method of Compensation, 

There are many cases in which a cause of error cannot 
conveniently be rendered null, and is yet beyond the 
reach of the third method, that of calculating the requisite 
correction from independent observations. The magnitude 

> Graham's Ch^mieal BeporU and Memoin, Cavendish Society, 
pp. 247, 268, &c 


of an error may be subject to continual variations, on 
account of change of weather, or other fickle cirumstances 
beyond our controL It may either be impracticable to 
observe the variation of those circumstances in sufficient 
detail, or, if observed, the calculation of the amount of 
error may be subject to doubt In these cases, and only 
in these cases, it will be desirable to invent some artificial 
mode of counterpoising the variable error against an equal 
error subject to exactly the same variation. 

We cannot weigh an object with great accuracy unless 
we make a correction for the weight of the air displaced 
by the object, and add this to the apparent weight In 
very accurate investigations relating to standard weights, 
it is usual to note the barometer and thermometer at the 
time of making a weighing, and, from the measured bulks 
of the objects compared, to calculate the weight of air 
displaced ; the third method in fact is adopted. To make* 
these calculations in the frequent weighings requisite in 
chemical analysis would be exceedingly laborious, hence 
the correction is usually neglected. But when the chemist 
wishes to weigh gas contained in a large glass globe for 
the purpose of determining it^ specific gravity, the correc- 
tion becomes of much importance. Hence chemists avoid 
at once the error, and the labour of correcting it, by 
attaching to the opposite scale of the balance a dummy 
sealed glass globe of equal capacity to that containing the 
gas to be weighed, noting only the difference of weight 
when the operating globe is full and empty. The correc- 
tion, being the same for both globes, may be entirely 

A device of nearly the' same kind is employed in the 
construction of galvanometers which measure the force of 
an electric current by the deflection of a suspended 
magnetic needle. The resistance of the needle is partly 
due to the directive influence of the earth's magnetism, 
and partly to the torsion of the thread. But the former 
force may often be inconveniently great as well as 
troublesome to determine for different inclinations. Hence 
it is customary to connect together two equally magnetised 
needles, with their poles pointing in opposite directions, 

* Rcgnanlt's Court EUmerUaire de ChimU, 185 1, vol i p. 141. 


I -A 










oiie needle being within and another without the coil of 
wire. As regards the earth's magnetism, the needles are 
now astatie or indifferent, the tendency of one needle 
towards the pole being balanced by that of the other. 

An elegant instance of the elimination of a disturbing 
force by compensation is found in Faraday's researches 
upon -the magnetism of gases. To observe the magnetic 
attraction or repulsion of a gas seems impossible unless we 
enclose the gas in an envelope, probably best made of 
glass. But any such envelope is sure to be more or less 
affected by the magnet, so that it becomes difficult to 
distinguish between three forces wliich enter into the 
problem, namely, the magnetism of the gas in question, 
that of the envelope, and that of the surrounding atmo- 
spheric air. Faraday avoided all difficulties by employing 
two equal and similar glass tubes connected together, and 
•so suspended from the arm of a torsion balance that the 
tubes were in similar parts of the magnetic field. One 
tube being filled with nitrogen and the other with oxygen, 
it was found that the oxygen seemed to be attracted and 
the nitrogen repelled. The suspending thread of the 
balance was then turned until the force of torsion restored 
the tubes to their original places, where the magnetism of 
the tubes as well as that of the sun-ounding air, being 
the same and in the opposite directions upon the two tubes, 
could not produce any interference. The force required 
to restore the tubes was measured by the amount of 
torsion of the thread, and it indicated correctly the dif- 
ference between the attractive powers of oxygen and 
nitrogen. The oxygen was then withdrawn from one of 
the tubes, and a second experiment made, so as to compare 
a vacuum with nitrogen. No force was now required to 
maintain the tubes in their places, so that nitrogen was 
found to be, approximately speaking, indifferent to the 
magnet, that is, neither magnetic nor diaraagnetic, while 
oxygen was proved to be positively magnetic.^ It required 
the highest experimental skill on the part of Faraday 
and TyndaD, to distinguish between what is apparent and 
real in magnetic attraction and repulsion. 

Experience alone can finally decide when a com- 

» Tyndall*8 Faraday ^ pp. 114, 115. 


pensating arrangement is conducive to accuracy. As a 
general rule mechanical compensation is the last resource, 
and in the more accurate observations it is likely to 
introduce more uncertainty than it removes. A multitude 
of instruments involving mechanical compensation have 
been devised, but they are usually of an unscientific 
character,^ because the errors compensated can be more 
accurately determined and allowed for. But there are 
exceptions to this rule, and it seems to be proved that in 
the delicate and tiresome operation of measuring a base 
line, invariable bars, compensated for expansion by heat, 
give ths most accurate results. This arises from the fact 
that it is very difiicult to determine accurately the 
temperature of the measuring bars under varying con- 
ditions of weather and manipulation.^ Again, the last 
refinement in the measurement of time at Greenwich 
Observatory depends upon mechanical compensation. Sir 
George Airy, observing that the standard clock increased, 
its losing rate 030 second for an increase of one inch in 
atmospheric pressure, placed a magnet moved by a baro- 
meter in such a position below the pendulum, as almost 
entirely to neutralise this cause of irregularity. The 
thorough remedy, however, would be to remove the cause 
of error altogether by placing the clock in a vacuous case. 
We thus see that the choice of one or other mode of 
eliminating an error depends entirely upon circumstances 
and the object in view ; but we may safely lay down the 
following conclusions. First of all, seek to avoid the 
source of error altogether if it can be conveniently done ; 
if not, make the experiment so that the error may be as 
small, but more especially as constant, as possible. If the 
means are at hand for determining its amount by calcula- 
tion from other experiments and principles of science, allow 
the error to exist and make a correction in the result K 
this cannot be accurately done or involves too much labour 
for the purposes in view, then throw in a counteracting 
error which shall as nearly as possible be of equal amount 
in all circumstances with that to be eliminated. There yet 
remains, however, one important method, that of Keversal, 

* Bee, for mstance, the Compensated Sympiesometer, PhUotophical 
Magaziney 4th Series, vol. xxxix. p. 371. 

* Grant, History of Phytical Asbrorwmy, pp. 146, 147. 

A A 





which will form an appropriate ti'ansition to the succeediog 
chapters on the Method of Mean Results and the Law of 


5. Method of Reversal, 

The fifth method of eliminating error is most potent 
and satisfactory when it can be applied, but it requires 
that we shall be able to reveree the apparatus and mode 
of procedure, so as to make the interfering cause act 
alternately in opposite directions. If we can get two 
experimental results, one of which is as much too great as 
the other is too small, the error is equal to half the dif- 
ference, and the true result is the mean of the two 
apparent results. It is an unavoidable defect of the 
chemical balance, for instance, that the points of suspen- 
sion of the pans cannot be fixed at exactly equal distances 
from the centre of suspension of the beam. Hence two 
weights which seem to balance each other will never be 
quite equal in reality. The difiference is detected by re- 
versing the weights, and it may be estimated by adding 
small weights to the deficient side to restore equilibrium, 
and then taking as the true weight the geometric mean of 
the two apparent weights of the same object If the 
difference is small, the arithmetic mean, that is half the 
sum, may be substituted for the geometric mean, from which 
it will not appreciably differ. 

This method of reversal is most extensively employed 
in practical astronomy. The apparent elevation of a 
heavenly body is observed by a telescope moving upon 
a divided circle, upon which the inclination of the 
telescope is read off". Now this reading will be erroneous 
if the circle and the telescope have not accurately the 
same centre. But if we read off* at the same time both 
ends of the telescope, the one reading will be about as 
much too small as the other is too great, and the mean 
will be nearly free from error. In practice the observa- 
tion is differently conducted, but the principle is the same ; 
the telescope is fixed to the circle, which moves with it, 
and the angle through which it moves is read off at three, 
six, or more points, disposed at equal intervals round the 
circle. The older astronomers, down even to the time oi 


Flamsteed, were accustomed to use portions only of a 
divided circle, generally quadrants, and Eomer made a 
vast improvement when he introduced the complete circle. 

The transit circle, employed to determine the meridian 
passiige of heavenly bodies, is so constmcted that the 
telescope and the axis bearing it, in fact the whole moving 
part of the instrument, can be taken out of the bearing 
sockets and turned over, so that what was formerly the 
western pivot becomes the eastern one, and vice versd. 
It is impossible that the instrument could have been 
80 perfectly constructed, mounted, and adjusted that the 
telescope should point exactly to the meridian, but the 
effect of the reversal is that it will point as much to 
the west in one position as it does to the east in the 
other, and the mean result of observations in the two 
positions must be free from such cause of error. 

The accuracy with which the inclination of the compass 
needle can be determined depends almost entirely on the 
method of reversal The dip needle consists of a bar 
of magnetised steel, suspended somewhat like the beam of 
a delicate balance on a slender axis passing through the 
centre of gravity of the bar, so that it is at liberty to rest 
in that exact degree of inclination in the magnetic meridian 
which the magnetism of the earth induces. The inclina- 
tion is read off upon a vertical divided circle, but to avoid 
error arising from the centring of the needle and circle, 
both ends are read, and the mean of the results is taken. 
The whole instrument is now turned carefully round 
through 180°, which causes the needle to assume a new 
position relatively to the' circle and gives two new readings, 
in which any error due to the wrong position of the zero 
of the division will be reversed. As the axis of the needle 
may not be exactly horizontal, it is now reversed in the 
same manner as the transit instrument, the end of the axis 
which formerly pointed east being made to point west, and 
a new set of four readings is taken. 

Finally, error may arise from the axis not passing 
accurately through the centre of gravity of the bar, and 
this error can only be detected and eliminated on chang- 
ing the magnetic poles of the bar by the application of a 
strong magnet. The error is thus made to act in opposite 
directions. To ensure all possible accuracy each reversal 

AA 2 



ought to be combined with each other reversal, so that the 
needle will be observed in eight diPTorent positions by 
sixteen readings, the mean of the whole of which will give 
the required inclination free from all eliminable errors.* 

There are certain cases in which a disturbing cause can 
with etise be made to act in opposite directions, in alter- 
nate observations, so that the mean of the results will be 
free from disturbance. Thus in direct experiments upon 
the velocity of sound in passing through the air between 
stations two or three miles apart, the wind is a cause of 
error. It will be well, in the first place, to choose a time 
for the experiment when the air is very nearly at rest, and 
the disturbance slight, but if at the same moment signal 
sounds be made at each station and observed at the other, 
two sounds will be passing in opposite dii-ections tlirough 
the same body of air and the wind will accelerate one 
sound almost exactly jis it retards the other. Again, in 
trigonometrical surveys the apparent height of a point will 
be affected by atmospheric refraction and the curvature of 
the earth. But if in the case of two points the apparent 
elevation of each as seen from the other be observed, the 
corrections will be the same in amount, but reversed in 
direction, and the mean between the two apparent dif- 
ferences of altitude will give the true difference of level. 

In the next two chapters we really pursue the Method 
of Keversal into more complicated applications. 

* Quetelet, Sur la Physique du Olobe, p. 174. Janiiu, Court lU 
Physique, vol. i. p. 504. 

, t 





All results of the measurement of continuous quantity 
1 can be only approximately true. Were this assertion 
doubted, it could readily be proved by direct experience. 
If any person, using an instrument of the greatest pre- 
cision, makes and registers successive observations in 
an unbiassed manner, it ,will almost invariably be found 
that the results differ from each other. When we operate 
with sufficient care we cannot perform so simple an 
experiment as weighing an object in a good balance 
without getting discrepant numbers. Only the rough 
and careless experimenter will think that his observations 
agree, but in reality he will be found to overlook the 
differences. The most elaborate researches, such as those 
undertaken in connection with standard weights and 
measures, always render.it apparent that complete coinci- 
dence is out of the question, and that the more accurate 
our modes of observation are rendered, the more numerous 
are the sources of minute error which become apparent. 
We may look upon the existence of error in all measure- 
^ ments as the normal state of things. It is absolutely 
' impossible to eliminate separately the multitude of small 
disturbing influences, except by balancing them off against 
each other. Even in drawing a mean it is to be expected 
that we shall come near the truth rather than exactly to 
it In the measurement of continuous quantity, absolute 
coincidence, if it seems to occur, must be only apparent, 
and is no indication of precision. It is one of the most 
embarrassing things we can meet when experimental 


ii a 





As restdts agree too closely. Such coincidences should raise 
! our suspicion that the apparatus in use is in some way 
restricted in its operation, so as not really to give the true 
result at all, or that the actual results have not heen faith- 
fully recorded by the assistant in charge of the apparatus. 
If then we cannot get twice over exactly the same 
result, the question arises, How can we ever attain the 
truth or select the result which may be supposed to 
approach most nearly to it ? The quantity of a certain 
phenomenon is expressed in several numbers which differ 
from each other ; no more tlian one of them at the most 
can be true, and it is more probable that they are all 
false. It may be suggested, perhaps, that the observer 
should select the one observation which he judged to be 
the best made, and there will often doubtless be a feeling 
that one or more results were siitisfactory, and the others 
less trustworthy. This seems to have been the course 
adopted by the early astronomers. Flamsteed, when he 
had made several observations of a star, probably chose in 
an arbitrary manner that which 'seemed to him nearest to 
the truth.^ 

When Horrocks selected for his estimate of the sun's 
semi-diameter a mean between the results of Kepler and 
Tycho, he professed not to do it from any regard to the 
idle adage, "Medio tutissimus ibis," but because he 
thought it from his own observations to be correct* But 
this method will not apply at all when the obsei-ver has 
made a number of measurements which are equally good 
in his opinion, and it is quite apparent that in using an 
instrument or apparatus of considerable complication the 
observer will not necessarily be able to judge whether 
slight causes have affected its operation or not. 

In this question, as indeed throughout inductive logic, 
we deal only with probabilities. There is no infallible 
mode of amving at the absolute truth, which lies beyond 
the reach of human intellect, and can only be the distant 
object of our long-continued and painful approximations. 
Nevertheless there is a mode pointed out alike by common 
sense and the highest mathematical reasoning, which is 

' Baily^s Account of Flamstscdy p. 376. 

' The TrantU of Vmut acrou the Sun^ by Horrock^ LondoD, 1859. 
p. 146. 




more likely than any other, as a general rule, to bring us 
near the truth. The apiarov ficTpov, or the aurea mediocritas, 
was highly esteemed in the ancient philosophy of Greece 
and Rome ; but it is not probable that any of the ancients 
should have been able clearly to analyse and express the 
reasons why they advocated the Toean as the safest course. 
But in the last two centuries this apparently simple 
question of the mean has been found to afford a field for 
the exercise of the utmost mathematical skill. Roger 
Cotes, the editor of the Principia, appears to have had 
some insight into the value of the mean ; but profound 
mathematicians such as De Moivre, Daniel Bernoulli, 
Laplace, Lagrange, Gauss, Quetelet, De Morgan, Airy, 
Leslie Ellis, Boole, Glaisher, and others, have hardly ex- 
hausted the subject 

Several uses of the Mean Result. 

The elimination of errors of unknown sources, is almost 
always accomplished by the simple arithmetical process 
of taking the mean, or, as it is often called, the average 
of several discrepant numbers. To take an average is to 
add the several quantities together, and divide by the 
number of quantities thus added, which gives a quotient 
lying among, or in the middle of, the several quantities. 
Before however inquiring fully into the grounds of this 
procedure, it is essential to observe that this one arith- 
metical process is really applied in at least three different 
cases, for different purposes, and upon different principles, 
and we must take great care not to confuse one applica- 
tion of the process with another. A m^n result, then, 
may have any one of the following significations. 

(1) It may give a merely representative number, 
expressing the general magnitude of a series of quantities, 
and serving as a convenient mode of comparing them 
with other series of quantities. Such a number is properly 
called Thefictitwus mean or The average result. 

(2) It may give a result approximately free from 
disturbing quantities, which are known to affect some 
results in one dii-ection, and other results equally in the 
opposite direction. We may say that in this case we get 
a Precise mean result. 








(3) It may give a result more or less free from imknown 
and uncertain errors; this we may call the Probable 
mean result. 

Of these three uses of the mean the first is entirely dif- 
ferent in nature from the two last, since it does not yield 
an approximation to any natural quantity, but furnishes 
us with an arithmetic result comparing the aggregate of 
certain quantities with their number. The third use of 
the mean rests entirely upon the theory of probability, 
and will be more fully considered in a later part of this 
chapter. The second use is closely connected, or even 
identical with, the Method of Keversal already described, 
but it will be desirable to enter somewhat fully into all the 
three employments of the same arithmetical process. 

7.%e Mean and the Average. 

Much confusion exists in the popular, or even the 
scientific employment of the terms mean and average, and 
they are commonly taken as synonymous. It is necessar}' 
to ascertain caiefully what significations we ought to 
attach to them. The English word m^an is equivalent to 
medium, being derived, perhaps through the French moijen, 
from the Latin mcdius, which again is undoubtedly kindred 
with the Greek fMcaofi. Etymologists l>elieve, too, that this 
Greek word is connected with the preposition fiera, the 
German miite, and the true English viid or middle ; so that 
after all the m^^an is a technical term identical in its root 
with the more popular equivalent middle. 

If we inquire what is the mean in a mathematical point 
of view, the true answer is that there are several or many 
kinds of means. The old arithmeticians recognised ten 
kinds, which are stated by Boethius, and an eleventh was 
added by Jordanus.* 

The arithmetic m^ean is the one by far the most 
commonly denoted by the term, and that which we may 
understand it to signify in the absence of any qualification. 
It is the sum of a series of quantities divided by their 
number, and may be represented by the formula i (a + b). 

' De Morgan, Sapplement to the Penny Oyelopadia, art. Old 
AppeUatimu of Nwnberg. 

But there is also the geometric m^an, which is the square 
root of the product, V<» X b, or that quantity the loga- 
rithm of which is the arithmetic mean of the logarithms 
of the quantities. There is also the harmonic mean, 
which is the reciprocal of the arithmetic mean of the 
reciprocals of the quantities. Thus if a and b be the 

quantities, as before, their reciprocals are - and r, the 
mean of which is ^ ^- 4. i), and the reciprocal again is 

— p-,, which is the harmonic mean. Other kinds of 

means might no doubt be invented for particular purposes, 
and we might apply the term, as De Morgan pointed 
out,^ to any quantity a function of which is equal to 
a function of two or more other quantities, and is such 
that the interchange of these latter quantities among them- 
selves will make no alteration in the value of the function. 

SymbolicaUy, if 4)(y,y,y ) = (pt^, «,, ajg . . . .), then y 

is a kind of mean of the quantities, Xi, x^, &c. 

The geometric mean is necessarily adopted in certain 
cases. When we estimate the work done against a force 
which varies inversely as the square of the distance from a 
fixed point, the mean force is the geometric mean between 
the forces at the beginning and end of the path. When in 
an imperfect balance, we reverse the weights to eliminate 
error, the true weight will be the geometric mean of the 
two apparent weights. In almost all the calculations of 
statistics and commerce the geometric mean ought, strictly 
speaking, to be used. If a commodity rises in price 100 
per cent, and another remains unaltered, the mean rise of 
a price is not 50 per cent, because the ratio 150 : 200 is 
n ot the same as 100 : 150. The mean ratio is as unity to 
s/roo X 200 or I to 1*41. The difference between the 
three kinds of means in such a case * is very considerable ; 
while the rise of price estimated by the Arithmetic mean 
would be 50 per cent, it would be only 41 and 33 per cent, 
respectively according to the Geometric and Harmonic 

' Penny Cydopcedia, art Mean. 

* Jevons, Journal of the Statistical Society f June 1865, ^^l' ^cxviii 
p. 296. 

I , 



In all calculations concerning the average rate of 
progress of a community, or any of its operations, the 
geometric mean should be employed. For if a quantity 
increases loo per cent, in loo years, it would not on the 
average increase lo per cent, in each ten years, as the 
lo per cent, would at the end of each decade be calculated 
upon larger and larger quantities, and give at the end of 
too years much more than loo per cent., in fact as much 
as 159 per ^ent. The true mean rate in each decade 
would be *J/2~ or about 107, that is, the increase would 
be about 7 per cent, in each ten years. But when the 
quantities differ very little, the arithmetic and geometric 
means are approximately the same. Thus the arithmetic 
mean of rcxxD and i 001 is i 0005, and the geometric mean 
is about I 0004998, the difference being of an oitler in- 
appreciable in almost all scientific and practical matters. 
Even in the comparison of standard weights by Gauss' 
method of reversal, the arithmetic mean may usually be 
substituted for the geometric mean which is the true result 

Regarding the mean in the absence of express qualifica- 
tion to the contrary as the common arithmetic mean, we 
must still distinguish between its two uses where it 
gives with more or less accuracy and probability a 
really existing quantity, and where it acts as a mere 
representative of other quantities. If I make many 
experiments to determine the atomic weight of an element, 
there is a certain immber which I wish to approximate to, 
and the mean of my separate results will, in the absence 
of any reasons to the contrary, be the most probable 
approximate result When we determine the mean 
density of the earth, it is not because any part of the earth 
is of that exact density ; there may be no part exactly 
corresponding to the mean density, and as the crust of tlie 
earth has only about half the mean density, the internal 
matter of the globe must of course be above the mean. 
Even the density of a homogeneous substance like carbon 
or gold must be regarded as a mean between the real 
density of its atoms, and the zero density of the interven- 
nig vacuous space. 

The very different signification of the word " mean " in 
these two uses was fully explained by Quetelet,^ and the 

* Liters <m iht Tluory of ProbabilUia, transl. by Downes, Part iL 




importance of the distinction was pointed out by Sir John 
Herschel in reviewing his work.^ It is much to be desired 
that scientific men would mark the difference by using the 
word mean only in the former sense when it denotes ap- 
proximation to a definite existing quantity ; and average, 
when the mean is only a fictitious quantity, used for con- 
venience of thought and expression. The etymology of 
this word " average " is somewhat obscure ; but according 
to De Morgan * it comes from aver-ia, " havings or pos- 
sessions," especially applied to farm stock. By the acci- 
dents of language averagium came to mean the labour of 
farm horses to which the lord was entitled, and it prob- 
ably acquired in this manner the notion of distributing a 
whole into parts, a sense in which it was early applied to 
maritime averages or contributions of the other owners of 
cargo to those whose goods have been thrown overboard or 
used for the safety of the vessel. 

On ilve Average or FicHtums Mean. 

Although the average when employed in its proper 
sense of a fictitious mean, represents no really existing 
quantity, it is yet of the highest scientific importance, as 
enabling us to conceive in a single result a multitude of 
details. It enables us to make a hypothetical simplifica- 
tion of a problem,and avoid complexity without committing 
error. The weight of a body is the sum of the weights of 
infinitely small particles, each acting at a different place, 
so that a mechanical problem resolves itself, strictly speak- 
ing, into an infinite number of distinct problems. We 
owe to Archimedes the first introduction of the beautiful 
idea that one point may be discovered in a gravitating 
body such that the weight of all the particles may be re- 
garded as concentrated in that point, and yet the behaviour 
of the whole body will be exactly represented by the 
behaviour of this heavy point This Centre of Gravity 
may be within the body, as in the case of a sphere, or it 
may be in empty space, as in the case of a ring. Any two 
bodies, whether comiected or separate, may be conceived 

1 HerscheVs Essaytf &c ppi 404, 405. 

* On the Theory of Errors of Observationtf Camhridge FhUotophical 
Transactions, yoL x. Part ii 416. 

< u 







, ^1 

V t ' 

as having a centre of gravity, that of the sun and earth 
lying within the sun and only 267 miles from its centre. 

Although we most commonly use the notion of a centre 
or average point with regard to gravity, the same notion 
is applicable to other cases. Terrestrial gravity is a case 
of approximately parallel forces, and the centre of gravity 
is but a special case of the more general Centre of Parallel 
Forces. Wherever a number of forces of whatever amount 
act in parallel lines, it is possible to discover a point at 
which the algebraic sum of the forces may be imagined to 
act with exactly the same effect Water in a cistern 
presses against the side with a pressure varying according 
to the depth, but always in a direction perpendicular to 
the side. We may then conceive the whole pressure as 
exerted on one point, which will be one-third from the 
bottom of the cistern, and may be called the Centre of 
Pressure. The Centre of Oscillation of a pendulum, dis- 
covered by Huyghens, is that point at which the whole 
weight of the pendulum may be considered as concentrated, 
without altering the time of oscillation (p. 315). When 
one body strikes another the Centre of Percussion is that 
point in the striking body at which all its mass might be 
concentrated without altering the effect of the stroke. In 
position the Centre of Percussion does not differ from the 
Centre of Oscillation. Mathematicians have also described 
the Centre of Gyration, the Centre of Convei-sion, the 
Centre of Friction, &c. 

We ought carefully to distinguish between those cases 
in which an invarialle centre can be assigned, and those in 
which it cannot. In perfect strictness, there is no such 
thing as a true invariable centre of gravity. As a general 
rule a body is capable of possessing an invariable centre 
only for perfectly parallel forces, and gravity never does 
act in absolutely parallel lines. Thus, as usual, we find 
that our conceptions are only hypothetically correct, and 
only approximately applicable to real circumstances. 
There are indeed certain geometrical forms called Centro- 
baric} such that a body of that shape would attract another 
exactly as if the mass were concentrated at the centre of 
gravity, whether the forces act in a parallel manner or not 

^ Thomson and Tait» Tr«U%H on NiUmnU Fhilotophy, voL i p. 594. 





Newton showed that uniform spheres of matter have this 
property, and this truth proved of the greatest importance 
in simplifying his calculations. But it is after all a purely 
hypothetical truth, because we can nowhere meet with, nor 
can we construct, a perfectly spherical and homogeneous 
body. The slightest iiTegularity or protrusion from the 
surface will destroy the rigorous correctness of the assump- 
tion. The spheroid, on the other hand, has no invariable 
centre at which its mass may always be regarded as con- 
centrated. The point from which its resultant attraction 
acts will move about according to the distance and posi- 
tion of the other attracting body, and it will only coincide 
with the centre as regards an infinitely distant body whose 
attractive forces may be considered as acting in parallel 

Physicists speak familiarly of the poles of a magnet, and 
the term may be used with convenience. But, if we attach 
any definite meaning to the word, the poles are not the 
ends of the magnet, nor any fixed points within, but the 
variable points from which the resultants of all the forces 
exerted by the particles in the bar upon exterior magnetic 
particles may be considered as acting. The poles are, in 
short. Centres of Magnetic Forces ; but as those forces are 
never really parallel, these centres will vary in position 
according to the relative place of the object attracted. 
Only when we regard the magnet as attracting a very 
distant, or, strictly speaking, infinitely distant particle, do 
its centres become fixed points, situated in short magnets 
approximately at one-sixth of the whole length from each 
end of the bar. We have in the above instances of centres 
or poles of force sufficient examples of the mode in which 
the Fictitious Mean or Average is employed in physical 

The Precise Mean Remit, 

We now turn to that mode of employing the mean 
result which is analogous to the method of reversal, but 
which is brought into practice in a most extensive manner 
throughout many branches of physical science. We find 
the simplest possible case in the determination of the lati- 
tude of a place by observations of the Pole-star. Tycho 






; f ■ 








Brahe suggested that if the elevation of any circumpolar 
star were observed at its higher and lower passages across 
the meridian, half the sum of the elevations would be tlie 
latitude of the place, which is equal to the height of the 
pole. Such a star is as much above the pole at its highest 
passage, as it is below at its lowest, so that the mean must 
necessarily give the height of the pole itself free from 
doubt, except as regards incidental errors. The Pole-star 
is usually selected for the purpose of such observations 
because it describes the smallest circle, and is thus on the 
whole least affected by atmospheric refraction. 

Whenever several causes are in action, each of which at 
one time increases and at another time decreases the joint 
effect by equal quantities, we may apply this method and 
disentangle the effects. Thus the solar and lunar tides 
roll on in almost complete independence of each other. 
When the moon is new or full the solar tide coincides, or 
nearly so, with that caused by the moon, and the joint 
effect is the sum of the separate effects. When the moon 
is in quadrature, or half full, the two tides are acting in 
opposition, one raising and the other depressing the water, 
so that we observe oi3y the difference of the effects. We 
have in fact — 

Spring tide = lunar tide + solar tide ; 

Neap tide = lunar tide — solar tide. 
We have only then to add together the heights of tlie 
maximum spring tide and the minimum neap tide, and 
half the sum is the true height of the lunar tide. Half 
the difference of the spring and neap tides on the other 
hand gives the solar tide. 

Effects of very small amount may be detected with 
great approach to certainty among much greater fluctua- 
tions, provided that we have a series of observations suf- 
ciently immerous and long continued to enable us to 
balance all the larger effects against each other. For this 
purpose the observations should be continued over at least 
one complete cycle, in which the effects run through all 
their variations, and return exactly to the same relative 
positions as at the commencement. If casual or irregular 
disturbing causes exist, we should probably require many 
such cycles of results to render their effect inappreciable. 
We obtain the desired result by taking the mean of all the 




observations in which a cause acts positively, and the 
mean of all in which it acts negatively. Half the diffe- 
I'ence of these means will give the effect of the cause in 
question, provided that no other effect happens to vary in 
tiie same period or nearly so. 

Since the moon causes a movement of the ocean, it is 
evident that its attraction must have some effect upon the 
atmosphere. The laws of atmospheric tides were investi- 
gated by Laplace, but as it would be impracticable by 
theory to calculate their amounts we can only determine 
them by observation, as Laplace predicted that they would 
one day be determined.^ But the oscillations of the 
barometer thus caused are far smaller than the oscillations 
due to several other causes. Storms, hurricanes, or changes 
of weather produce movements of the barometer some- 
times as much as a thousand times as great as the tides in 
question. There are also regular daily, yearly, or other 
fluctuations, all greater than the desired quantity. To 
detect and measure the atmospheric tide it was desirable 
that observations should be made in a place as free as 
possible from irregular disturbances. On this account 
several long series of observations were made at St. 
Helena, where the barometer is far more regular in its 
movements than in a continental clinrnte. The effect of 
the moon's attraction was then detected by taking the 
mean of all the readings when the moon was on the me- 
ridian and the similar mean when she was on the horizon. 
The difference of these means was found to be only 
00^6$, yet it was possible to discover even the variation 
of this tide according as the moon was nearer to or further 
from the earth, though this difference was only 00056 
inch.* It is -quite evident that such minute effects could 
never be discovered in a purely empirical manner. Having 
no information but the series of observations before us, 
we could have no clue as to the mode of grouping them 
which would give so small a difference. In applying this 
method of means in an extensive manner we must gene- 
rally then have d priori knowledge as to the periods at 
which a cause will act in one direction or the other. 

* E»»a% PhUosophiqiie sur lei ProbabiliUs, pp. 49, 50. 

• Grant, Hitiory of Physical Astronomy, p. 163. 












We are sometimes able to eliminate fluctuations and 
take a mean result by purely mechanical an-angements. 
The daily variations of temperature, for instance, become 
imperceptible one or two feet below the surface of the 
earth, so that a thermometer placed with its bulb at that 
depth gives very nearly the true daily mean temperature. 
At a depth of twenty feet even the yearly fluctuations are 
nearly eflaced, and the thermometer stands a little above 
the true mean temperature of the locality. In registering 
the rise and fall of the tide by a tide-gauge, it is desirable 
to avoid the oscillations arising from surface waves, which 
is very readily accomplished by placing the float in a cis- 
tern communicating by a small hole with the sea. Only a 
general rise or fall of the level is then perceptible, just as 
in the marine barometer the narrow tube prevents any 
casual fluctuations and allows only a continued change of 
pressure to manifest itself. 

Determination of the Zero point. 

In many important observations the chief difficulty con- 
sists in defining exactly the zero point from which we are 
to measure. We can point a telescope with great pre- 
cision to a star and can measure to a second of arc the 
angle through which the telescope is raised or lowered ; 
but all this precision will be useless unless we know 
exactly the centre point of the heavens from which we 
measure, or, what comes to the same thing, the horizontal 
line 90° distant from it. Since the true horizon has 
reference to the figure of the earth at the place of 
observation, we can only determine it by the direction 
of gravity, as marked either by the plumb-line or the 
surface of a liquid. The question resolves itself then into 
the most accurate mode of observing the direction of 
gravity, and as the plumb-line has long been found 
hopelessly inaccurate, astronomers generally employ the 
surface of mercury in repose as the criterion of horizon- 
tality. They ingeniously observe the direction of the 
surface by making a star the index. From the laws 
of reflection it follows that the angle between the 
direct ray from a star and that reflected from a surface 
of mercury will be exactly double the angle between the 




surface and the direct ray from the star. Hence the 
horizontal or zero point is the mean between the apparent 
place of any star or other very distant object and its 
reflection in mercury. 

A plumb-line is perpendicular, or a liquid surface is 
horizontal only in an approximate sense ; for any irregu- 
larity of the surface of the earth, a mountain, or even a 
house must cause some deviation by its attracting power. 
To detect such deviation might seem very difficult, because 
every other plumb-line or liquid surface would be equally 
affected by gravity. Nevertheless it can be detected ; for 
if we place one plumb-line to the north of a mountain, and 
another to the south, they will be about equally deflected 
in opposite directions, and if by observations of the same 
star we can measure the angle between the plumb-lines, 
half the inclination will be the deviation of either, after 
allowance has been made for the inclination due to the 
difference of latitude of the two places of observation. By 
this mode of observation applied to the mountain Schiehal- 
lion the deviation of the plumb-line was accurately measured 
by Maskelyne, and thus a comparison instituted between 
the attractive forces of the mountain and the whole globe, 
which led to a probable estimate of the earth's density. 

In some cases it is actuaUy better to determine the zero 
point by the average of equally diverging quantities than 
by direct observation. In delicate weighmgs by a chemical 
balance it is requisite to ascertain exactly the point at 
which the beam comes to rest, and when standard weights 
are being compared the position of the beam is ascertained 
by a carefully divided sciale viewed through a microscope. 
But when the beam is just coming to rest, friction, small 
impediments or other accidental causes may readily ob- 
struct it, because it is near the point at which the force of 
stability becomes infinitely small Hence it is found better 
to let the beam vibrate and observe the terminal points of 
the vibrations. The mean between two extreme points 
will nearly indicate the position of rest Friction and 
the resistance of air tend to reduce the vibrations, so that 
this mean will be eiToneous by half the amount oi this 
effect during a half vibratioa But by taking several ob- 
servations we may determine this retardation and allow 
for it Thus if a, ft, c be the readings of the terminal 

B B 



■ 1 1 








points of three excursions of the beam from the zero of the 
scale, then J (a + ft) will be about as much erroneous in 
one direction as ^ (ft + c) in the other, so that the mean 
of these two means, or J (a + 2 ft + c), will be exceedingly 
near to the point of rest^ A still closer approximation 
may be made by taking four readings and reducing them 
by the formula J(a + 2ft4-2 c '\- d). 

The accuracy of Baily's experiments, directed to deter- 
mine the density of the earth, entirely depended upon this 
mode of observing oscillations. The balls whose gmvi- 
tation was measured were so delicately suspended by a 
torsion balance that they never came to rest The extreme 
points of the oscillations were observed both when the 
heavy leaden attracting ball was on one side and on the 
other. The difference of the mean points when the leaden 
ball was on the right hand and that when it was on the 
left hand gave double the amount of the deflection. 

A beautiful instance of avoiding the use of a zero point 
is found in Mr. K J. Stone's observations on the radiant 
heat of the fixed stars. The difficulty of these obsei-vations 
arose from the comparatively great amounts of heat which 
were sent into the telescope from the atmosphere, and which 
were sufficient to disguise almost entirely the feeble heat 
rays of a star. But Mr. Stone fixed at the focus of his 
telescope a double thermo-electric pile of which the two 
parts were reversed in order. Now any disturbance of 
temperature which acted uniformly upon both piles pro- 
duced no effect upon the galvanometer needle, and when 
the rays of the star were made to fall alternately upon 
one pile and the other, the total amount of the deflection 
represented double the heating power of the star. Thus 
Mr. Stone was able to detect with much certainty a heating 
effect of the star Arcturus, which even when concentrated 
by the telescope amounted only to o°02 Fahr., and which 
represents a heating effect of the direct ray of only about 
o°ocxxx)i37 Fahr., equivalent to the heat which would be 
received from a three-inch cubic vessel full of boiling 
water at the distance of 400 yards.* It is probable that 

* Gaues, Taylor's ScierUi/ic Memoirt, vol. ii. p. 43, &c. 

* Proeudings of the Moyal Society t vol. xviii. p. 159 (Jan. 13, 1870). 
Pkilosophical Magaziru (4th Series), voL xxxix. p. 376. 




Mr. Stone's arrangement of the pile might be usefully 
employed in other delicate thermometric experiments 
subject to considerable disturbing influences. 

Determination of Maximum Points. 

We employ the method of means in a certain number 
of observations directed to determine the moment at which 
a phenomenon reaches its highest point in quantity. In 
noting the place of a fixed star at a given time there is no 
difficulty in ascertaining the point to be observed, for a 
star in a good telescope presents an exceedingly small disc. 
In observing a nebulous body which from a bright centre 
fades gradually away on all sides, it will not be possible 
to select with certainty the middle point. In many such 
cases the best method is not to select arbitrarily the sup- 
posed middle point, but points of equal brightness on 
either side, and then take the mean of the observations of 
these two points for the centre. As a general rule, a 
variable quantity in reaching its maximum increases at a 
less and less rate, and after passing the highest point 
begins to decrease by insensible degrees. The maximum 
may indeed be defined as that point at which the increase 
or decrease is null. Hence it will usually be the most 
indefinite point, and if we can accurately measure the 
phenomenon we shall best determine the place of the 
maximum by determining points on either side at which 
the ordinates are equal. There is moreover this advantage 
in the method that several points may be determined with 
the corresponding ones on the other side, and the mean of 
the whole taken as the true place of the maximum. But 
this method entirely depends upon the existence of sym- 
metry in the curve, so that of two equal ordinates one 
shall be as far on one side of the maximum as the other 
is on the other side. The method fails when other laws of 
variation prevail 

In tidal observations great difficulty is encountered in 
fixing the moment of high water, because the rate at which 
the water is then rising or falling, is almost impercep- 
tible. Whewell proposed, therefore, to note the time at 
which the water passes a fixed point somewhat below the 
maximum both in rising; and falling, and take the mean 

BB 2 













time as that of high water. But this mode of proceeding 
unfortunately does not give a correct result, because the 
tide follows different laws in rising and in falling. There 
is a difficulty again in selecting the highest spring tide, 
another object of much importance in tidology. Laplace 
discovered that the tide of the second day preceding the 
conjunction of the sun and moon is nearly equal to that of 
the fifth day following; and, believing that the increase 
and decrease of the tides proceeded in a nearly symmetrical 
manner, he decided that the highest tide would occur about 
thirty-six hours after the conjunction, that is half-way 
between the second day before and the fifth day after.* 

This method is also emplo}'ed in determining the time 
of passage of the middle or densest point of a stream of 
meteors. The earth takes two or three days in passing 
completely through the November stream ; but astronomers 
need for their calculations to have some definite point fixed 
within a few minutes if possible. When near to the 
middle they observe the numbers of meteors which come 
within the sphere of vision in each half hour, or quartei 
hour, and then, assuming that the law of variation is 
symmetrical, they select a moment for the passage of the 
centre equidistant between times of equal frequency. 

The eclipses of Jupiter's satellites are not only of great 
interest as regards the motions of the satellites themselves, 
but were, and perhaps still are, of use in determining 
longitudes, because they are events occurring at fixed 
moments of absolute time, and visible in all parts of the 
planetary system at the same time, allowance being made 
for the interval occupied by the light in travelling. But, 
as is explained by Herschel,* the moment of the event is 
wanting in definiteness, partly because the long cone of 
Jupiter's shadow is surrounded by a penumbra, and partly 
because the satellite has itself a sensible disc, and takes 
time in entering the shadow. Different obseiTers using 
different telescopes would usually select different moments 
for that of the eclipse. But the increase of light in the 
emersion will proceed according to a law the reverse of 
that observed in the immersion, so that if an observer notes 

' Airy On Tides and Waves, Encycl. Metrop. pp. 364* — 366*. 
* OuiUnu qf Astronomy^ 4th edition, { 538 




the time of both events with the same telescope, he will be 
as much too soon in one observation as he is too late in the 
other, and the mean moment of the two observations will 
represent with considerable accuracy the time when the 
satellite is in the middle of the shadow. Error of judg- 
ment of the observer is thus eliminated, provided that 
he takes care to act at the emei-sion as he did at the 





\ h 

/ ( 

To bring error itself under law might seem beyond human 
power. He who errs surely diverges from law, and it 
might be deemed hopeless out of error to draw truth. One 
of the most remarkable achievements of the human intel- 
lect is the establishment of a general theory which not only 
enables us among discrepant results to approximate to 
the truth, but to assign the degree of probability which 
fairly attaches to this conclusion. It would be a mistake 
indeed to suppose tliat this law is necessarily the best 
guide under all circumstances. Every measuring instru- 
ment and every form of experiment may have its own 
special law of error ; there may in one instrument be a 
tendency in one direction and in another in the opposite 
direction. Every process has its peculiar liabilities to 

disturbance, and we are never i-elieved from the necessity of 
providing against special difficulties. The general Law of 
Enx)r is the best guide only when we have exhausted all 
other means of approximation, and still find discrepancies, 
which are due to unknown causes. We must treat such 
residual differences in some way or other, since they wiU 
^ occur in all accurate experiments, and as their origin is 
assumed to be unknown, there is no reason why we should 
treat them differently in different cases. Accordingly the 
ultimate Law of Error must be a uniform and general one. 
It is perfectly recognised by mathematicians that in 
each case a special Law of Error may exist, and should be 
discovered if possible. "Nothing can be more unlikely 
t^an that the errors committed in all classes of observa- 

Cll. XVII.] 



tions should follow the same law," ^ and the special Laws 
of Error which will apply to certain instmments, as for in- 
stance the repeating circle, have been investigated by 
Bi*avais.2 He concludes that every distinct cause of error 
gives rise to a curve of possibility of errors, which may 
have any form, — a curve which we may either be able or 
unable to discover, and which in the first case may be 
determined by d priori considerations on the peculiar 
nature of this cause, or which may be determined d 
posteriori by observation. Whenever it is practicable and 
worth the labour, we ought to investigate these special 
conditions of error ; nevertheless, when there are a great 
number of different sources of minute error, the general 
resultant will always tend to obey that general law which 
we are about to consider. 

Establishment of the Law of Erroi: 

Mathematicians agree far better as to the form of the 
Law of Error than they do as to the manner in which it 
can be deduced and proved. They agree that among a 
number of discrepant results of observation, that mean 
quantity is probably the best approximation to the truth 
which makes the sum of the squares of the errors as small 
as possible. But there are three prin cipaLway s i n which 
this ^ law J iasJ^gen^arriveH" at Tespgctive^,l)y Grauss^ by 
LajJace^'Sid^.uetelet.^SSIE^ ^ir'Jo I m Herscbel. Gauss 
proceeds^uch upon assumptions^ Herschel rests*" upon 
geometrical considerations ; while Laplace and Quetelet 
regard the Law of Error as a development of the doctrine 
of combinations. A number of other mathematicians, such 
as Adrain of New Brunswick, Bessol, Ivory, Donkin, Leslie 
Ellis, Tait, and Crofton have either attempted independent 
proofs or have modified or commented on those here to be 
described. For full accounts of the literature of the 
subject the reader should refer either to Mr. Todhunter's 
History of the Theory of Prohahility or to the able memoir 
of Mr. J. W. L Glaisher.8 

' Philosophical Magazine^ 3rd Series, vol xxxvii. p. 324. 

* Letters on the Theory of FrohabilitieSy by Quetelet, translated by 
0. G. Downes, Notes to Letter XXVL pp. 286—295. 

^ On the Law of Facility of Errors of OhservationSj and on th* 
Method of Least Squares^ Memoirs of the Royal Astroaomical Society, 
Yol. xxxix. p. 75. 

^'1 1 








According to Gauss the Law of Error expresses the 
comparative probability of errors of various magnitude, and 
partly from experience, partly from d ^^rixyri considera- 
tions, we may readily lay down certain conditions to which 
the law will certainly conform. It may fairly be assumed 
as a first principle to guide us in the selection of the 
law, that large errors will be far less frequent and probable 
than small ones. We know that very large errors are 
almost impossible, so that the probability must rapidly 
decrease as the amount of the error increases. A second 
principle is that positive and n^ative errors shall be 
equally probable, which may certainly be assumed, because 
we are supposed to be devoid of any knowledge as to the 
causes of the residual errors. It follows that the proba- 
bility of the error must be a function of an even power of 
the magnitude, that is of the square, or the fourth power, 
or the sixth power, otherwise the probability of the same 
amount of error would vary according as the error was 
positive or negative. The even powers ai", a?*, ««, &c., are 
always intrinsically positive, whether x be positive or 
negativa There is no <i ^grwri reason why one rather than 
another of these even powers should be selected. Gauss 
himself allows that the fourth or sixth power would fulfil 
the conditions as well as the second ; * but in the absence 
of any theoretical reasons we shoultl prefer the second 
power, because it leads to formulae of great comparative 
simplicity. Did the Law of Error necessitate the use of 
the higher powers of the error, the complexity of the 
necessary calculations would much reduce the utility of 
the theory. 

By mathematical reasoning which it would be unde- 
sirable to attempt to follow in this book, it is shown 
that under these conditions, the facility of occurrence, 
or in other words, the probability of error is expressed 
by a function of the general form c"** •", in which x repre- 
sents the variable amount of errors. From this law, 
to be more fully described in the following sections, it at 
once follows that the most probable result of any observa- 

» Mithfidt dea Moindres Carrit, Mivwire* tur la CombimnUon dei 
OWohoiM, par Ch. Fr Oauss. Ttaduit m Fran^aU par J. 
9«r<rani, Pans, 1855, pp. 6, 133, &c ^ V ^' 




tions is that which makes the sum of the squares of 
the consequent errors the least possible. Let a, h, c, 
&C., be the results of observation, and x the quantity 
selected as the most probable, that is the most free 
from unknown errors : then we must determine x so that 
(a - ac)* + (J - »)* + (c - a;)2 + . . . . shall be the least 
possible quantity. Thus we arrive at the celebrated 
MdhoiJL^^f-^LMst Sqvures, as it is usually called, which 
appears to have been first distinctly put in practice by 
Gauss in 1795, while Legendre first published in 1806 an 
account of the process in his work, entitled, Nouvelles 
Mithodes pour la Determination des Orhites des CorrUtes. It 
is worthy of notice, however, that Roger Cotes had long 
previously recommended a method of equivalent nature in 
his tract. " Estimatio Erroris in Mixta Mathesl" ^ 

Her8chel*8 Geometrical Proof, 

A second way of arriving at the Law of Error was 
proposed by Herschel, and although only applicable to 
geometrical cases, it is remarkable as showing that from 
whatever point of view we regard the subject, the same 
principle will be detected. Aft/Cr assuming that some 
general law must exist, and that it is subject to the 
principles of probability, hfe supposes that a ball is 
dropped from a high point with the intention that it 
shall strike a given mark on a horizontal plane. In the 
absence of any known causes of deviation it will either 
strike that mark, or, as is infinitely more probable, diverge 
from it by an amount which we must regard as error of 
unknown origin. Now, to quote the words of Herschel,^ 
" the probability of that error is the unknown function of 
its square, i.e. of the sum of the squares of its deviations in 
any two rectangular directions. Now, the probability of 
any deviation depending solely on its magnitude, and not 
on its direction, it follows that the probability of each of 
these rectangular deviations must be the same function of 
its square. And since the observed oblique deviation is 

* De Morgan, Penny Cyclopaedia, art. Least Squares, 

* Edinburgh Bevitu), July 1850, vol. xciL p. 17. Reprinted EsgaySy 
p. 399. This method of demonstration is discussed by Boole, Tram- 
Qdiom of Royal Society qf Edinburgh^ voL xxi. pp. 627 — 630, 


! 1 






equivalent to the two rectaDgular ones, supposed concur- 
rent, and which are essentially independent of one another, 
and is, therefore, a compound event of which they are the 
simple independent constituents, therefore its probability 
will be the product of their separate probabilities. Thus 
the form of our unknown function comes to be determined 
from this condition, viz., that the product of such functions 
of two independent elements is equal to the same function 
of their sum. But it is shown in every work on algebra 
that this property is the peculiar characteristic of, and 
belongs only to, the exponential or antilogarithmic function. . 
This, then, is the function of the square of the error, which 
vf expresses the probability of committing that error. That 
probability decreases, therefore, in geometrical progression, 
as the square of the error increases in arithmetical." 

Laplace s and Queteht*8 Proof of the Law, 

However much presumption the modes of determining 
the Law of Error, already described, may give in favour of 
the law usually adopted, it is difficult to feel that the 
arguments are satisfactory. The law adopted is chosen 
rather on the grounds of convenience and pkusibDity, than 
because it can be seen to be the necessary law. We can 
however approach the subject from an entirely different 
point of view, and yet get to the same result. 

Let us assume that a particular observation is subject 
to four chances of error, each of which will increase the 
result one inch if it occurs. Each of these errors is to be 
regarded as an event independent of the rest and we can 
therefore assign, by the theory of probability, the compara- 
tive probability and frequency of ea^h conjunction of errors. 
From the Arithmetical Triangle (pp. 182-188) we learn that 
no error at all can happen only in one way ; an error of 
one inch can happen in 4 ways ; and the ways of happening 
of errors of 2, 3 and 4 inches respectively, will be 6, 4 and 
I in number. 

We may infer that the error of two inches is the most 
likely to occur, and will occur in the long run in six cases 
out of sixteen. Errors of one and three inches will be 
equally likely, but will occur less frequently ; while no 
eiTor at all or one of four inches will be a comparatively 


> — , — 

rare occurrence. If we now suppose the errors to act as 
often in one direction as the other, the effect will be to 
alter the average error by the amount of two inches, and 
we shall have the following results : — 

Negative error of 2 inches i way 

Negative error of i inch 4 ways. 

No error at all 6 ways. 

Positive error of i inch ...... 4 ways. 

Positive error of 2 inches i way. 

We may now imagine the number of causes of error 
increased and the amount of each error decreased, and the 
aiithmetical triangle will give us the frequency of the re- 
sulting errors. Thus if there be five positive causes of 
error and five negative causes, the following table shows 
the numbers of errors of various amount which will be the 
result : — 

Direction of Error. 

Positive Error. 

Negative Error. 

Amount uf EUror. 

5. 4. 3. a. « 


i> 3> 3> 4. 5 

Number of such ESrrors. 

I, 10, 45, 120, 210 

aio, lao, 45, lo, i 

It is plain that from such numbers I can ascertain 
the probability of any particular amount of en'or under 
the conditions supposed. The probability of a positive 


error of exactly one inph is > in which fraction the 

•^ 1024 

numerator is the number of combinations giving one 
inch positive error, and the denominator the whole 
number of possible errors of all magnitudes. I can also, 
by adding together the appropriate numbers get the pro- 
bability of an error not exceeding a certain amount. Thus 
the probability of an error of three inches or less, positive 
or negative, is a fraction whose numerator is the sum of 
45 4- 120 -f- 210 4- 252 + 210 + 120 -{■ 45, and the deno- 

minator, as before, giving the result 



We may see at 

once that, according to these principles, the probability of 
small errors is far greater than of large ones : the odds are 
1002 to 22t or more than 45 to i that the error will not 










exceed three inches ; and the odds are 1022 to 2 against 
the occurrence of the greatest possible error of five inches. 
If any case should arise in which the observer knows 
the number and magnitude of the chief errors which 
may occur, he ought certainly to calculate from the Arith- 
metical Triangle the special Law of Error which would 
apply. But the general law, of which we are in search, 
is to be used in the dark, when we have no knowledge 
whatever of the sources of error. To assume any special 
number of causes of error is then an arbitrary pi-oceeding, 
. rand mathematicians have chosen the least arbitrary course 
' 'ol imagining the existence of an infinite number of in- 
|fibDitely small errors, just as, in the inverse method of 
/probabilities, an infinite number of infinitely improbable 
'hypotheses were submitted to calculation (p. 255). 

The reasons in favour of this choice are of several 
different kinds. 

1. It cannot be denied that there may exist infinitely 
numerous causes of error in any act of observation. 

2. The law resulting from the hypothesis of a moderate 
number of causes of error, does not appreciably differ from 
that given by the hypothesis of an infiiute number of 
causes of error. 

3. We gain by the hypothesis of infinity a general law 
capable of ready calculation, and applicable by uniform 
rules to all problems. 

4. This law, when tested by comparison with extensive 
series of observations, is strikingly verified, as will be 
shown in a later section. 

When we imagine the existence of any large number of 
causes of error, for instance one hundred, the numbers of 
combinations become impracticably large, as may be seen 
to be the case from a glance at the Arithmetical Triangle, 
which proceeds only up to the seventeenth line. Quetelet, 
by suitable abbreviating processes, calculated out a table 
of probability of errors on the hypothesis of one thousand 
distinct causes;* but mathematicians have generally 
proceeded on the hypothesis of infinity, and then, by the 
devices of analysis, have substituted a general law of easy 

» Ldters on the Theory of ProbdbiliHet, Letter XV. and Appendix, 
aote pp. 256 -266. 





treatment. In mathematical works upon the subject, it is 
shown that the standard Law of Error is expressed in the 

in which x is the amount of the error, Y the maximum 
ordinat'C of the curve of error, and c a number constant 
for each series of observations, and expressing the amount 
of the tendency to error, varying between one series of 
observations and another. The letter e is the mathematical 
constant, the sum of ratios between the numbers of permu- 
tations and combinations, previously referred to (p. 330). 

To show the close correspondence of this general 
law with the special law which might be derived 
from the supposition of a moderate number of causes 
of error, I have in the accompanying figure drawn a 

-5 -I -B -I -1 -t 

curved line representing accurately the variation of y 

when X in the above formula is taken equal o, -, i, - 2, 

&c., positive or negative, the arbitmry quantit^s Y and c 
being each assumed equal to unity, in order to simplify 
the calculations. In the same figure are inserted eleven 
dots, whose heights above the base line are proportional 
to tJie numbers in the eleventh line of the Arithmetical 
Triangle, thus representing the comparative probabilities 
of errors of various amounts arising tix>m ten equal causes 










of error. The correspondence of the general and the 
special Law of Error is almost as close as can be exhibited 
in the figure, and the assumption of a greater number of 
equal causes of error would render the correspondence far 
more close. 

It may be explained that the ordinates NM, nw, n'm\ 
represent values of y in the equation expressing the I^aw 
of Error. The occurrence of any one definite amount of 
error is infinitely improbable, because an infinite number 
of such ordinates might l)e drawn. But the probability of 
an error occuiTing between certain limits is finite, and is 
represented by a portion of the area of the curve. Thus the 
probability that an error, positive or negative, not exceed- 
ing unity will occur, is represented by the area Mmnn'm', 
in short, by the area standing upon the line nn. 
Since every observation must either have some definite 
error or none at all, it follows that the whole area of the 
curve should be considered as the unit expressing certainty, 
and the probability of an error falling between particular 
limits will then be expressed by the ratio which the area 
of the curve between those limits bears to the whole area 
of the curve. 

The mere fact that the Law of Error allows of the posst 
ble existence of qtyots of every assignable amount showa 
that it is only approximately true. We may fairly say 
that in measuring a mile it would be impossible to commit 
an error of a hundred miles, and the length of life would 
never allow of our committing an error of one million 
miles. Nevertheless the general Law of Error would assign 
a probability for an error of that amount or more, but so 
small a probability as to be utterly inconsiderable and 
almost inconceivable. All that can, or in fact need, be 
said in defence of the law is, that it may be made to re- 
present the errors in any special case to a very close 
approximation, and that the probability of large and prac- 
tically impossible errors, as given by the law, will be so 
small as to be entirely inconsiderable. And as we are 
dealing with error itself, and our results pretend to nothing 
more than approximation and probability, an indefinitely 
small error in our process of approximation is of no import- 
ance whatever. 

xtii I 



Logical Origin of the Lata of Error, 

It is worthy of notice that this Law of Error, abstruse 
though the subject may seem, is really founded upon the 
simplest principles. It arises entirely out of the difference 
between permutations and combinations, a subject upon 
which I may seem to have dwelt with unnecessary prolixity 
in previous pages (pp. 170, 189). The order in which we 
add quantities together does not affect the amount of the 
sum, so that if there be three positive and five negative 
causes of error in operation, it does not matter in which 
order they are considered as acting. They may be inter- 
mixed in any arrangement, and yet the result will be the 
same. The reader should not fail to notice how laws or 
principles which appeared to be absurdly simple and 
evident when first noticed, reappear in the most complicated 
and mysterious processes of scientific method. The funda- 
mental Laws of Identity and Difference gave rise to the 
Logical Alphabet which, after abstracting the character of 
the differences, led to the Arithmetical Triangle. Th6 
Law of Error is defined by an infinitely high line of that 
triangle, and the law proves that the mean is the most pro- 
bable result, and that divergencies from the mean become 
much less probable as they increase in amount. Now the 
comparative greatness of the numbers towards the middle 
of each line of the- Arithmetical Triangle is entirely due 
to the indifference of order in space or time, which was 
first prominently pointed out as a condition of logical re- 
lations, and the symbols indicating them (pp. 32-35), and 
which was afterwards shown to attach equally to numerical 
symbols, the derivatives of logical terms (p. 160). 

Verification of the Law of Error. 

The theory of error which we have been considering 
rests entirely upon an assumption, namely that when 
known sources of disturbances are allowed for, there yet 
remain an indefinite, possibly an infinite number of other 
minute sources of error, which will as often produce ex- 
cess as deficiency. Granting this assumption, the Law of 
Error must be as it is usually taken to be, and there is 
DO more need to verify it empirically than to test the truth 











of one of Euclid's propositions mechanically. Neverthe- 
less, it is an interesting occupation to verify even the pro- 
positions of geometry, and it is still more instructive to 
try whether a large number of observations will justify our 
assumption of the Law of Error. 

Encke has given an excellent instance of the correspond- 
ence of theory with experience, in the case of observations 
of the differences of Right Ascension of the sun and two 
stars, namely a Aquilse and a Canis minoris. The obser- 
vations were 470 in number, and were made by Bradley 
and reduced by Bessel, who found the probable error of 
the final result to be only about one-fourth part of a second 
(0*2637). He then compared the numbers of errors of 
each magnitude from o* i second upwards, as actually given 
by the observations, with what should occur according to 
the Iaw of Error. 

The results were as follow : — ^ 

of a Moond. 

Namber of erron of each magnitado 
Moording to 



00 to o't 
I ., •« 

• .. -3 
'3 - '4 

♦ - *i 
•5 ., « 

7 f» 

•8 ,. 9 

-9 M « .» 

» bore < 








The reader will remark that the correspondencif is very 
close, except as regards larger errors, which are excessive 
in practica It is one objection, indeed, to the theory of 
error, that, being expressed in a continuous mathematical 
function, it contemplates the existence of errors of every 
magnitude, such as could not practically occur ; yet in this 
case the theory seems to under-estimate the number of 
large errors. 

1 Encke, On the Method 0/ Lead iSgitarei, Taylor's ScietU^fic 
Mumoinf vol ii. pp. 338, 339. 




Another comparison of the law with observation was made 
by Quetelet, who investigated the errors of 487 determi- 
nations in time of the Right Ascension of the Pole-Star 
made at Greenwich during the four years 1836-39. These 
observations, although carefully corrected for all known 
causes of error, as well as for nutation, precession, &c., 
are yet of course found to differ, and being classified as 
regards intervals of one-half second of time, and then pro- 
portionately increased in number, so that their sum may 
be one thousand, give the following results as compared 
with what Quetelet's theory would lead us to expect : — * 

Magnitude of 

error in truths 

of a second. 

Number of Errors 

Magnitude of 

error in tcntlis 

of 4 second. 

Number of errors 






+ 05 
+ ro 

+ 1-5 
+ a 

+ •5 
+ 3-0 












— I'O 

-I 5 
— 20 



— 35 








In this instance also the correspondence is satisfactory, 
but the divergence between theory and fact is in the opposite 
direction to that discovered in the former comparison, the 
larger errors being less frequent than theory would indi- 
cate. It will be noticed that Quetelet's theoretical results 
are not symmetrical 

The Probable Mean Result. 

One immediate result of the Law of Error, as thus 
stated, is that the mean result is the most probable one ; 
and when there is only a single variable this mean is 
found by the familiar arithmetical process. An unfor- 
tunate error has crept into several works which allude 
to this subject. Mill, in treating of the " Elimination of 
Chance,*' remarks in a note * that " the mean is spoken of 

' Quetelet, Letters on the Tluory of Probabilities, translated by 
Downes, Letter XIX. p. 88. See also Galtoii's Ilereditary Oenxus, 

p. ^79. 

System of Logic, bk. iiL chap. 17, § 3. 5tb ed. vol. ii. p. 56. 

G C 













as if it were exactly the same thing as the average. 
But the mean, for purposes of inductive inquiry, is not the 
average, or arithmetical mean, though in a familiar illus- 
tration of the theory the difference may be disregarded.** 
He goes on to say that, according to mathematical princi- 
ples, the most probable result is that for which the sums 
of the squares of the deviations is the least possible. It 
seems probable that Mill and other writers were misled 
by Whewell, who says^ that "Tlie method of least 
squares is in fact a method of means, but with some 
peculiar characters. . . . The method proceeds upon 
this supposition : that all errors are not equally probable, 
but that small errors are more probable than large ones." 
He adds that this method '* removes much that is arbitrary 
in the method of means." It is strange to find a mathe- 
matician like Whewell making such remarks, when there 
is no doubt whatever that the Method of Means is only 
an application of the Method of Least Squares. They are, 
in fact, the same method, except that the latter method 
may be applied to cases where two or more quantities have 
to be determined at the same time. Lubbock and Drink- 
water say,* " If only one quantity has to be determined, 
this method evidently resolves itself into taking the mean 
of all the values given by observation." Encke says,' that 
the expression for the probability of an error " not only 
contains in itself the principle of the arithmetical mean, 
but depends so immediately upon it, that for all those 
magnitudes for which the arithmetical mean holds good 
in the simple cases in which it is principally applied, 
no other law of probability can be assumed than that 
which is expressed by this formula." 

The Probahle Error of Results. 

When we draw a conclusion from the numerical 
results of observations we ought not to consider it suf- 
ficient, in cases of importance, to content ourselves with 
finding the simple mean and treating it as true. We 
ought also to ascertain what is the degree of confidence 

* Pkiloiophy of the Inductive Sciences, 2nd ed. vol. ii. pp. 408, 409. 

* Euay <m FrohahUity, Useful Knowledge Society, 1833, p. 41. 

* Taylor's Scientific Memoiri^ vol. ii. p. 333. 




we may place in this mean, and our confidence should be 
measured by the degree of concurrence of the observations 
from which it is derived. In some cases the mean may 
be approximately certain and accurate. In other cases it 
may really be worth little or nothing. The Law of Error 
enables us to give exact expression to the degree of con- 
fidence proi)er in any case ; for it shows how to calculate 
the probability of a divergence of any amount from the 
mean, and we can thence ascertain the probability that 
the mean in question is within a certain distance from the 
true number. The probable error is taken by mathema- 
ticians to mean the limits within which it is as likely as 
not that the truth will fall. Thus if 5 45 be the mean of 
all the determinations of the density of the earth, and '20 
be approximately the probable error, the meaning is that 
the probability of the real density of the earth falling be- 
tween 5 2 5 and 5 65 is J. Any other limits might have 
been selected at will. We might calculate the limits 
within which it was one hundred or one thousand to one 
that the truth would fall ; but there is a convention to 
take the even odds one to one, as the quantity of proba- 
bility of which the limits are to be estimated. 

Many books on probability give rules for making the 
calculations, but as, in the progress of science, persons 
ought to become more familiar with these processes, 
I propose to repeat the rules here and illustrate their 
use. The calculations, when made in accordance with 
the directions, involve none but arithmetic or logar- 
ithmic operations. 

The following are the rules for treating a mean result, 
so as thoroughly to ascertain its trustworthiness. 

1. Draw the mean of all the observed results. 

2. Find the excess or defect, that is, the error of each 
result from the mean. 

3. Square each of these reputed errors. 

4. Add together all these squares of the errors, which 
are of course all positive. 

5. Divide by one less than the number of observations. 
This gives the sqiuire of the mean error, 

6. Take the square root of the last result ; it is the mean 
error of a single observation. 

7. Divide now by the square root of tlie number of 

cc 2 









observations, and we get the mean error of tJie mean 

8. Lastly, multiply by the natural constant 06745 (or 
approximately by 0*674, or even by J), and we arrive at 
the probable error of the mean result 

Suppose, for instance, that five measurements of the 
height of a hill, by the barometer or otherwise, have given 
the numbers of feet as 293, 301, 306, 307, 313 ; we want 
to know the probable error of the mean, namely 304. Now 
the differences between this mean and the above numbers, 
paying no regard to direction, are ii, 3, 2, 3, 9; their 
squares are 121, 9, 4, 9, 81, and the sum of the squares 
of the errors consequently 224. The number of observa- 
tions being 5, we divide by i less, or 4, getting 56. This 
is the square of the mean error, and taking its square root 
we have 7*48 (say 7 J), the mean error of a single obser- 
vation. Dividing by 2236, the square root of 5, the 
number of observations, we find the mean error of the mean 
result to be 3*35, or say 3 J, and lastly, multiplying by 
•6745, we arrive at the probable error of the mean result, 
which is found to be 2259, or say 2J. The meaning of 
this is that the probability is one half, or the odds are 
even that the true height of the mountain lies between 
301} and 306J feet. We have thus an exact measure of 
the degree of credibility of our mean result, which mean 
indicates the most likely point for the truth to fall 

The reader should observe that as the object in these 
calculations is only to gain a notion of the degree of con- 
fidence with which we view the mean, there is no real use 
in caiTying the calculations to any great degree of pre- 
cision ; and whenever the neglect of decimal fractions, or 
6ven the slight alteration of a number, will much abbre- 
viate the computations, it may be fearlessly done, except 
in cases of high importance and precision. Brodie has 
shown how the law of error may be usefully applied in 
chemical investigations, and some illustrations of its em- 
ployment may be found in his paper.* 

The experiments of Benzenberg to detect the revolution 
of the earth, by the deviation of a ball from the perpen- 

' Philosophical Tranaactwns, 1873, P* ^3* 




dicular line in falling down a deep pit, have been cited by 
Encke^ as an interesting illustration of the Law of Error. 
The mean deviation was 5 086 lines, and its probable error 
was calculated by Encke to be not more than -950 line, 
that is, the odds were even that the true result lay between 
4" 1 36 and 6036. As the deviation, according to astrono- 
mical theory, should be 46 lines, which lies well within 
the limits, we may consider that the experiments are 
consistent with the Copemican system of the universe. 

It will of course be understood that the probable error 
has regard only to those causes of errors which in the lonf« 
run act as much in one direction as another ; it takes no 
account of constant errors. The true result accordingly 
will often fall far beyond the limits of probable error, owing 
to some considerable constant error or errors, of the ex- 
istence of which we are unaware. 

Bisection of the Mean Besult. 

We ought always to bear in mind that the mean of any 
series of observations is the best, that is, the most probable 
approximation to the truth, only in the absence of know-s 
ledge to the contrary. The selection of the mean restst 
entirely upon the probability that unknown causes of eiTor 
will in the long run fall as often in one direction as the 
opposite, so that in drawing the mean they will balance 
each other. If we have any reason to suppose that there 
exists a tendency to error in one direction rather than the 
other, then to choose the mean would be to iguore that 
tendency. We may certainly approximate to the length 
of the circumference of a circle, by taking the mean of the 
perimeters of inscribed and circumscribed polygons of an 
equal and large number of sides. The length of the cir- 
cular line undoubtedly lies between the lengths of the two 
perimeters, but it does not follow that the mean is the 
best approximation. It may in fact be shown that the 
circumference of the circle is very nearly equal to the 
perimeter of the inscribed polygon, together with one -third 
part of the difference between the inscribed and circum- 
scribed polygons of the same number of sides. Having 

• Taylor's Scimtxfie Memoirt, vol. ii pp. 330, 347, &c. 


I I 1 

'•il.i . 


t '. 




this knowledge, we ought of course to act upon it, instead 
of trusting to probability. 

We may often perceive that a series of measurements 
tends towards an extreme limit rather than towards a 
mean. In endeavouring to obtain a correct estimate 
of the apparent diameter of tlie brightest fixed stars, we 
find a continuous diminution in estimates as the powers 
of observation increased. Kepler assigned to Sirius an 
apparent diameter of 240 seconds ; Tycho Brahe made 
it 126; Gassendi 10 seconds; Galileo, Hevelius, and J. 
Cassini, 5 or 6 seconds. Halley, Michell, and subsequently 
Sir W. Herscliel came to the conclusion that the brightest 
stars in the heavens could not have real discs of a second, 
and were probably much less in diameter. It would of 
course be absurd to take the mean of quantities which 
differ more than 240 times; and as the tendency has 
always been to smaller estimates, there is a considerable 
presumption in favour of the smallest.^ 

In many experiments and measurements we know that 
there is a preponderating tendency to error in one direc- 
tion. The readings of a thermometer tend to rise as 
the age of the instrument increases, and no drawing of 
means will correct this result. Barometers, on the other 
hand, are likely to read too low instead of too high, 
o\nng to the imperfection of the vacuum and the action of 
capillary attractioa If the mercury be perfectly pure and 
no appreciable enor be due to the measuring apparatus, 
the best barometer will be that which gives the highest 
result. In determining the specific gravity of a solid 
body the chief danger of error arises from bubbles of air 
adhering to the body, which would tend to make the 
specific gravity too small Much attention must always 
be given to one-sided errors of this kind, since the multi- 
plication of experiments does not remove the error. In 
such cases one very careful experiment is better than any 
number of careless ones. 

When we have reasonable grounds for supposing that 
certain experimental results are liable to grave errors, we 
should exclude them in drawing a mean. If we want to 
find the most probable approximation to the velocity of 

' Quetelet, Littert, &e. p. 1 16. 


THE LAW 01*' EltliOK. 


sound in air, it would be absurd to go back to the old 
experiments which made the velocity from 1200 to 1474 
feet per second ; for we know that the old observers did 
not guard against errors arising from wind and other 
causes. Old chemical experiments are valueless as re- 
gards quantitative results. The old chemists found the 
atmospliere in different places to differ in composition 
nearly ten per cent., whereas modern accurate experi- 
menters find very slight variations. Any method of 
measurement which we know to avoid a source of error 
is far to be preferred to others which trust to probabilities 
for the elimination of the eiTor. As Flamsteed says,^ " One 
good instrument is of as much worth as a hundred in- 
different ones." But an instrument is good or bad only in 
a comparative sense, and no instrument gives invariable 
and truthful results. Hence we must always ultimately 
fall back upon probabilities for the selection of the final 
mean, when other precautions are exhausted. 

Legendre, the discoverer of the method of Least Squares, 
recommended that observations differing very much from 
the results of his method should be rejected. The subject 
has been carefully investigated by Professor Pierce, who has 
proposed a criterion for the rejection of doubtful observa- 
tions based on the following principle:' — observations 
should be rejected when the probability of the system of 
errors obtained by retaining them is less than that of the 
system of errors obtained by their rejection multiplied by 
the probability of making so many and no more abnormal 
observations." Professor Pierce's investigation is given 
nearly in his own words in Professor W. Chauvenet's 
'* Manual of Spherical and Practical Astronomy," which 
contains a full and excellent discussion of the methods of 
treating numerical observations.^ 

Very difficult questions sometimes arise when one or 
more results of a method of experiment diverge widely 
from the mean of the rest. Are we or are we not to ex- 
clude them in adopting the supposed true mean result of 
the method? The drawing of a mean result rests, as I 

' Bailv, Account of Flamsiudy p. 56. 

' Gould's Atironomical Journal^ Cambridffe, Mass., vol. ii. p. 161. 

• Philadelphia (London, Triibner) 1863. Appendix, vol. ii. p. 558. 




I ! 





have frequently explained, upon the assumption that eveiy 
eiTor acting in one direction will probably be balanced by 
other errore acting in an opposite direction. If then we 
know or can possibly discover any causes of error not 
agreeing with this assumption, we shall be justified in 
excluding results which seem to be affected by this cause. 

In reducing large series of astronomical observations, it is 
not uncommon to meet with numbers diflfering from othei*s 
by a whole degree or half a degree, or some considerable in- 
tegral quantity. These are errors which could hardly arise 
in the act of observation or in instrumental irregularity ; 
but they might readily be accounted for by misreading 
of figures or mistaking of division marks. It would be 
absurd to trust to chance that such mistakes would 
balance each other in the long run, and it is therefore better 
to correct arbitrarily the supposed mistake, or better still, 
if new observations can be made, to strike out the diver- 
gent numbers altogether. When results come sometimes 
too great or too small in a regular manner, we should 
suspect that some part of the instrument slips through a 
definite space, or that a definite cause of error enters at 
times, and not at others. We should then make it a point 
of prime importance to discover the exact nature and 
amount of such an error, and either prevent its occurrence 
for the future or else introduce a corresponding correction. 
In many researches the whole difficulty will consist in 
this detection and avoidance of sources of error. Professor 
Hoscoe found that the presence of phosphorus caused 
serious and almost unavoidable enors in the determination 
of the atomic weight of vanadium.^ Herschel, in reducing 
his observations of double stars at the Cape of Good Hope,* 
was perplexed by an unaccountable difference of the angles 
of position as measured by the seven-feet equatorial and 
the twenty-feet reflector telescopes, and after a careful in- 
vestigation was obliged to be contented with introducing 
a correction experimentally determined.^ 

When observations are sufficiently numerous it seems 
desirable to project the apparent errors into a curve, and 
then to observe whether this curve exhibits the symmet- 

1 Bakeriaii Lecture, PhiloMphical Trantactions (1868), vol. clviiL 
p. 6. 
' Results of Observati&M ai the Cape of Oood Hope, p. 283. 





rical and characteristic form of the curve of error. If so, 
it may be inferred that the errors arise from many minute 
independent sources, and probably compensate each other 
in the mean result. Any considerable irregularity will 
indicate the existence of one-sided or large causes of error, 
which should be made the subject of investigation. 

Even the most patient and exhaustive investigations 
will sometimes fail to disclose any reason why some 
results diverge from others. The question again recurs — 
Are we arbitrarily to exclude them ? The answer should 
be in the negative as a general rule. The mere fact of 
divergence ought not to be taken as conclusive against a 
result, and the exertion of arbitrary choice would open 
the way to the fatal influence of bias, and what is com- 
monly known as the "cooking" of figures. It would 
amount to judging fact by theory instead of theory by fact. 
The apparently divergent number may prove in time to be 
the true one. It may be an exception of that valuable 
kind which upsets our false theories, a real exception, 
exploding apparent coincidences, and opening a way to a 
new view of the subject To establish this position for 
the divergent fact will require additional research ; but 
in the meantime we should give it some weight in our 
mean conclusions, and should bear in mind the discrepancy 
as one demanding attention. To neglect a divergent result 
is to neglect the possible clue to a great discovery. 

Method of Least Squares. 

When two or more unknown quantities are so involved 
that they cannot be separately determined by the Simple 
Method of Means, we can yet obtain their most probable 
values by the Method of Least Squares, without more 
difficulty than arises from the length of the arithmetical 
computations. If the result of each observation gives an 
equation between two unknown quantities of the form 

ax -\- by =■ c 
then, if the observations were free from error, we should 
need only two observations giving two equations; but for 
the attainment of greater accuracy, we may take many ol)- 
seryations, and reduce the equations so as to give only a 
pair with mean coefficients. This reduction is effected by 










(i.), multiplying the coefficients of eacli equation by the 
first coefficient, and adding togetlier all the similar co- 
efficients thus resulting for the coefficients of a new 
equation ; and (2.), by repeating this process, and multi- 
plying tlie coefficients of each equation by the coeflicient 
of the second term. Meaning by (sum of a*) the sum of 
all quantities of the same kind, and having the same place 
in the equations as a-, we may briefly describe the two 
resulting mean equations as follows : — 

(sum of a*) . aj + (sum of ad) . y = (sum of ac), 
(sum of ab) . X -\- (sum of i«) . y = (sum of be). 
When there are three or more unknown quantities 
the process is exactly the same in nature, and we get 
additional mean equations by multiplying by the third, 
fourth, &c., coefficients. As the numbers are in any case 
approximate, it is usually unnecessary to make the com- 
putations with accuracy, and places of decimals may be 
freely cut off to save arithmetical work. The mean 
equations having been computed, their solution by the 
ordinary methods of algebra gives the most probable 
values of the unknown quantities. 

Works upon the Theory of ProbabUUy, 

Regarding the Theory of Probability and the Law of 
Error a s most important subjects of s ^^dy fnr any onft who 
desi resHx) obtain a compl e te compre benaioQ of scientific 
me tEod as actually ^applied in physical investigations, I 
will briefly indicafc^tne works in one or other of which 
the reader ^m^best "pursue the study. 

The best popular, and at the same time profound English 
work on the subject is De Morgan's "Essay on Proba- 
bilities and on their Application to Life Contingencies and 
Insurance Offices," published in the Gahirui Cyclopctdia, 
and to be obtained (in print) from Messrs. Longman. 
Mr. Venn's work on The Logic of Chance can now be 
procured in a greatly enlarged second edition ; * it contains 
a moat interesting and able discussion of the metaphysical 

^ Thi Logic of Chance, an Essay on the FoundationB and Province 
of the Theory of Probability, with especial reference to its Logical 
Bearings and its Application to Moral and Social Science. (Mae> 
niiUau), 1876. 

basis of probability and of related questions concerning 
causation, belief, design, testimony, &c. ; but I cannot 
always agree with Mr. Venn's opinions. No mathematical 
knowledge beyond that of common arithmetic is required 
in reading these works. Quetelet's Letters form a good 
introduction to the subject, and the mathematical notes 
are of value. Sir George Airy's brief treatise On tlve 
Algebraical and Numerical Theory of Errors of Observa- 
tions and the Combination of Observations^ contains a 
complete explanation of the Law of Error and its prac- 
tical applications. De Morgan's treatise " On the Theory 
of Probabilities" in the Encyclopa^ia Metropolitana, 
presents an abstract oi the more abstruse investigations 
of Laplace, together with a multitude of profound and 
original remarks concerning the theory generally. In 
Lubbock and Drinkwater's work on Probability, in the 
Library of Useful Knowledge, we have a concise but 
good statement of a number of important problems. The 
Rev. W. A. Whitworth has given, in a work entitled 
Choice and Chance, a number of good illustrations of 
calculations both in combinations and probabilities. In 
Mr. Todhunter's admirable History we have an exhaustive 
critical account of almost all writings upon the subject of 
probability down to the culmination of the theory in 
I^place's works. The Memoir of Mr. J. W. L. Glaisher 
has already been mentioned (p. 375). In spite of the 
existence of these and some other good English works, 
there seems to be a want of an easy and yet pretty com- 
plete mathematical introduction to the study of the theory. 
Among French works the TraitS Elim^ntaire du Calcul 
des ProbabilUes, by S. F. Lacroix, of which several editions 
have been published, and which is not difficult to obtain, 
forms probably the best elementary treatise. Poisson's 
liecherches sur la FrobabilitS des Jugem&nts (Paris 1837), 
commence with an admirable investigation of the grounds 
and methods of the theory. While iSiplace's great Theorie 
Analytique des Frobabilitds is of course the " Principia " 
of the subject ; his Essai PhUosophiqu^e sur les Probability 
is a popular discourse, and is one of the most profound 
and interesting essays ever published. It should bo 
familiar to every student of logical method, and has lost 
little or none of its importance by lapse of time. 








Beleclion of Constant Errm's. 

The Method of Means is absolutely incapable of elimi- 
nating any error which is always the same, or which always 
lies in one direction. We sometimes require to be roused 
from a false feeling of security, and to be ui-ged to take 
suitable precautions against sucli occult errors. " It is 
to the observer," says Gauss,^ " that belongs the task of 
carefully removing the causes of constant errors," and this 
is quite true when the error is absolutely constant. When 
we have made a number of determinations with a certain 
apparatus or method of measurement, there is a great 
advantage in altering the arrangement, or even devising 
some entirely different method of getting estimates of the 
same quantity. The reason obviously consists in the im- 
probability that the same error will affect two or more 
different methods of experiment. If a discrepancy is 
found to exist, we shall at least be aware of the existence 
of error, and can take measures for finding in which way 
it lies. If we can try a considerable number of methods, 
the probabiUty becomes great that errors constant in one 
method will be balanced or nearly so by errors of an op- 
posite effect in the others. Suppose that there be thi-ee 
different methods each affected by an error of equal 
amount. The probability that this error will in all fall in 
the same direction is only J ; and with four methods 
similarly J. If each method be affected, as is always 
the case, by several independent sources of error, the 
probability becomes much gi-eater that in the mean result 
of all the methods some of the errora will partially 
compensate the others. In this case as in all others, when 
human vigilance has exhausted itself, we must trust the 
theory of probability. 

In the determination of a zero point, of the magnitude 
of the fundamental standards of time and space, in the 
personal equation of an astronomical observer, we have 
instances of fixed errors ; but as a general rule a change of 
procedure is likely to reverse the character of the error, 
and many instances may be given of the value of this 
precaution. If we measure over and over again the same 

^ Qauas, translated by Beitiuud, p. a(. 




angular magnitude by the same divided circle, maintained 
in exactly the same position, it is evident that the same 
mark in the circle will be the criterion in each case, and 
any error in the position of that mark will equally affect 
all our results. But if in each measurement we use a 
different part of the circle, a new mark will come into use, 
and as the error of each mark cannot be in the same 
direction, the average result will be nearly free from 
errors of division. It will be better still to use more 
than one divided circle. 

Even when we have no perception of the points at 
which error is likely to enter, we may with advantage 
vary the construction of our apparatus in the hope that we 
shall accidentally detect some latent cause of error. Baily's 
purpose in repeating the experiments of Michell and Caven- 
dish on the density of the earth was not merely to follow 
the same course and verify the previous numbers, but to 
try whether variations in the size and substance of the 
attracting balls, the mode of suspension, the temperature 
of the surrounding air, &c., would yield different results. 
He performed no less than 62 distinct series, comprising 
2153 experiments, and he carefully classified and discussed 
the results so as to disclose the utmost differences. Again, 
in experimenting upon the resistance of the air to the 
motion of a pendulum, Baily employed no less than 80 
pendulums of various forms and materials, in order to 
ascertain exactly upon what conditions the resistance 
depends. Regnault, in his exact researches upon the 
dilatation of gases, made arbitrary changes in the magni- 
tude of parts of his apparatus. He thinks that if, in spite 
of such modification, the results are unchanged, the errors 
are probably of inconsiderable amount ; ^ but in reality it 
is always possible, and usually likely, that we overlook 
sources of error which a future generation will detect. 
Thus the pendulum experiments of Baily and Sabine were 
directed to ascertain the nature and amount of a correction 
for air resistance, which had been entirely misunderstood 
in the experiments by means of the seconds pendulum, 
upon which was founded the definition of the standard 
yard, in the Act of 5th George IV. c. 74. It has already 

* Jamin, Couts de Physique, vol. ii. p. 60. 



been mentioned that a considerable error was discovered 
in the detennination of the standard metre as the ten- 
millionth part of the distance from the pole to the 
equator (p. 3145. 

We shall return in Chapter XXV. to the further consi- 
deration of the methods by which we may as far as possible 
secure ourselves against permanent and undetected sources 
of error. In the meantime, having completed the con- 
sideration of the special methods requisite for treating 
quantitative phenomena, we must pursue our principal 
subject, and endeavour to trace out the course by which 
the physicist, from observation and experiment, collects 
the materials of knowledge, and then proceeds by hypo- 
thesis and inverse calcmation to induce fix)m them the 
laws of nature. 





All knowledge proceeds originally from experience. Using 
J the name in a wide sense, we may say that experience 
comprehends all that we ftd, externally or internally^ 
the aggregate of the impressions which we receive through 
the various apertures of perception — the aggregate con- 
sequently of what is in the mind, except so far as some 
portions of knowledge may be the reasoned equivalents of 
other portions. As the word experience expresses, we go 
throtigh much in life, and the impressions gathered inten- 
tionally or unintentionally afford the materials from which 
the active powers of the mind evolve science. 

No small part of the experience actually employed in 
science is acquired without any distinct purpose. We 
cannot use the eye:^ without gathering some facts which 
may prove useful. A great science has in many cases 
risen from an accidental observation. Erasmus Bartholinus 
thus first discovered double refraction in Iceland spar; 
Galvani noticed the twitching of a frog's leg; Oken was 
struck by the form of a vertebra; Mains accidentally 
examined light reflected from distant windows with a 


' '1 





i ) 



double refracting substance ; and Sir John Hei-schel's 
attention was drawn to the peculiar appearance of a 
solution of quinine sulphate. In earlier times there must 
have been some one who first noticed the strange behaviour 
of a loadstone, or the unaccountable motions produced by 
amber. As a general rule we shall not know in what 
direction to look for a great body of phenomena widely 
different from those familiar to us. Chance then must 
give us the starting point ; but one accidental observation 
well used may lead us to make thousands of observations 
in an intentional and organised manner, and thus a science 
may be gradually worked out from the smallest opening. 

Distinction of Observation and Sxperi/runt. 

It is usual to say that the two sources of experience 
are Observation and Experiment. When we merely note 
and record the phenomena which occur around us in the 
^Ci)rdinary course of nature we are said to observe. When we 
change the course of nature by the intervention of our 
nmscular powers, and thus produce unusual combinations 
and conditions of phenomena, we are said to experiment. 
Herschel justly remarked ^ that we might properly call 
these two modes of experience passive and active observa- 
tion. In both cases we must certainly employ our senses 
to observe, and an experiment difiers from a mere observar 
tion in the fact that we more or less influence the 
character of the events which we observe. Experiment is 

^ thus observation plus alteration of conditions. 

It may readily be seen that we pass upwards by in- 
sensible gradations from pure observation to determinate 
experiment When the earliest astronomers simply noticed 
the ordinary motions of the sun, moon, and planets upon 
the face of the starry heavens, they were pure obseiTers. 

<^ But astronomers now select precise times and places for 
important observations of stellar parallax, or the transits 
of planets. They make the earth's orbit the basis of a 
.well arranged natural experivient, as it were, and take well 
considered advantage of motions which they cannot 
controL Meteorology might seem to be a science of pure 

' Preliminary Difcourse on the Study of Naimral Pkiloiophy, p. 77. 




observation, because we cannot possibly govern the changes 
of weather which we record. Nevertheless we may ascend 
mountains or rise in balloons, like Gay-Lussac and Glaisher, 
and may thus so vary the points of observation as to render 
our procedure experimental. We are wholly unable either 
to produce or prevent earth-currents of electricity, but 
when we construct long lines of telegraph, we gather such 
strong currents during periods of disturbance as to render 
them capable of easy observation. 

The best arranged systems of observation, however, would 
fail to give us a large part of the facts which we now 
possess. Many processes continually going on in nature 
are so slow and gentle as to escape our powers of observa- 
tion. Lavoisier remarked that the decomposition of water 
must have been constantly proceeding in nature, although 
its possibility was unknown till his time.^ No substance 
is wholly destitute of magnetic or diamagnetic powers ; 
but it required all the experimental skill of Faraday to 
prove that iron and a few other metals had no monopoly ^.^^ 
of these powers. Accidental observation long ago im-^y 
pressed upon men's minds the phenomena of lightning, 
and the attractive properties of amber. Experiment only 
could have shown that phenomena so diverse in magnitude 
and character were manifestations of the same agent. To 
observe with accuracy and convenience we must have 
agents under our control, so as to raise or lower their 
intensity, to stop or set diem in action at will. Just as 
Smeaton found it requisite to create an artificial and 
governable supply of wind for his investigation of wind- 
mills, so we must have governable supplies of light, heat, 
electricity, muscular force, or whatever other agents we are 

It is hardly needful to point out too that on the earth's 
surface we live under nearly constant conditions of gravity, 
temperature, and atmospheric pressure, so that if we are to 
extend our inferences to other parts of the universe where 
conditions are widely diflTerent, we must be prepared to 
imitate those conditions on a small scale here. We must 
have intensely high and low temperatures ; we must vary 


< • ;. 

* Layoisier's EUmmtt of Chemistry , translated by Kerr, 3rd ed. 
p. 148. 

n D 





the density of gases from approximate vacuum upwards ; 
we must subject liquids aud solids to pressures or stmins 
of almost unlimited amount. 

Mental Conditions of Correct Observation, 

Every observation must in a cei-tain sense be true, foi 
the observing and recording of an event is in itself an 
event But before we proceed to deal with the supposed 
meaning of the record, and dmw inferences concerning the 
course of nature, we must take care to ascertain that the 
character and feelings of the observer are not to a great 
extent the phenomena recorded. The mind of man, as 
Francis Bacon said, is like an uneven mirror, and does not 
reflect the events of nature without distortion. We need 
hardly take notice of intentionally false observations, nor 
of mistakes arising from defective memory, deficient light, 
and so forth. Even whei*e the utmost fidelity and care 
are used in observing and recoixling, tendencies to error 
exist, and fallacious opinions arise in consequence. 

It is difficult to find persons who can with perfect fair- 
ness register facts for and against their own peculiar views. 
Among uncultivated observers the tendency to remark 
favourable and foi-get unfavourable events is so great, that 
no reliance can be placed upon their supposed observations. 
Thus arises the enduring fallacy that the changes of the 
weather coincide in some way with the changes of the 
moon, although exact and impartial registei*s give no 
countenance to the fact. The whole race of prophets and 
quacks live on the overwhelming eftfect of one success, 
compared with hundreds of failures wliich are unmen- 
tioned and forgotten. As Bacon says, ** Men mark when 
they hit, and never mark when they miss." And we 
should do well to bear in mind the ancient story, quoted 
by Bacon, of one who in Pagan times was shown a temple 
with a picture of all the persons who had been saved from 
shipwreck, after paying their vows. When asked whether 
he did not now acknowledge the power of the gods, 
"Ay," he answered; "but where are they painted that 
were drowned after their vows ? " 

If indeed we could estimate the amount of bias existing 
in any particular observations, it might be treated like 




one of the forces of the problem, and the true course of 
external nature might still be rendered apparent. But the 
feelings of an observer are usually too indeterminate, so 

^ that when there is reason to suspect considerable bias, re- 
jection is the only safe course. As regards facts casually 
registered in past times, the capacity and impartiality of 
the observer are so little known that we shoiid spare no 
pains to replace these statements by a new appeal to 
nature. An indiscriminate medley of truth and absurdity, 
such as Francis Bacon collected in his Natural History, is 
wholly unsuited to the purposes of science. But of course 
when records relate to past events like eclipses, con- 
junctions, meteoric phenomena, earthquakes, volcanic 
eruptions, changes of sea margins, the existence of now 
extinct animals, the migrations of tribes, remarkable 
customs, &c., we must make use of statements however 
unsatisfactory, and must endeavour to verifiy them by the 
comparison of independent records or traditions. 

When extensive series of observations have to be made, 
as in astronomical, meteorological, or magnetical observa- 
tories, trigonometrical surveys, and extensive chemical or 
physical researches, it is an advantage that the numerical 
work should be executed by assistants who are not interested 

^ in, and are perhaps unaware of, the expected results. The 
record is thus rendered perfectly impartial. It may even 
be desirable that those who perform the purely routine 
work of measurement and computation should be un- 
acquainted with the principles of the subject. The great 
table of logarithms of the French Revolutionary (jovem- 
ment was worked out by a staff of sixty or eighty 
computers, most of whom were acquainted only with the 
rules of arithmetic, and worked under the direction of 
skilled mathematicians ; yet their calculations were usually 
found more correct than those of persons more deeply 
t/ versed in mathematics.^ In the Indian Ordnance Survey 
the actual measurers were selected so that they should 
not have sufficient skill to falsify their results without 

Both passive observation and experimentation must, 
however, be generally conducted by persons who know for 

^ Babbage, Economy nf Many/ticturetf p. 194. 

D D 2 





what they are to look. It is only when excited and guided 
by the hope of verifying a theory that the observer will 
notice many of the most important points ; and, where the 
work is not of a routine character, no assistant can super- 
sede the mind-direct-ed observations of the philosopher. 
Thus the successful investigator must combine diverse 
qualities ; he must have clear notions of the result he ex- 
pects and confidence in the truth of his theories, and yet 
he must have that candour and flexibility of mind which 
enable him to accept unfavourable results and abandon 
mistaken views. 

Instrumental and Sensual Condttums of Observation. 

In every observation one or more of the senses must be 
employed, and we should ever bear in mind that the ex- 
tent of our knowledge may be limited by the power of the 
sense concerned. What we learn of the world only forms 
the lower limit of what is to be learned, and, for all that 
we can tell, the processes of nature may infinitely sur- 
pass in variety and complexity those which are capable of 
coming within our means of observation. In some cases 
inference from observed phenomena may make us in- 
directly aware of what cannot be directly felt, but we 
can never be sure that we thus acquire any appreciable 
fraction of the knowledge that might be acquired. 

It is a strange reflection that space may be filled with 
dark wandering stars, whose existence could not have yet 
become in any way known to us. The planets have 
already cooled so far as to be no longer luminous, and it 
may well be that other stellar bodies of various size have 
fallen into the same condition. From the consideration, 
indeed, of variable and extinguished stars, Laplace inferred 
that there probably exist opaque bodies as great and 
perhaps as numerous as those we see.^ Some of these 
dark stars might ultimately become known to us, either 
by reflecting light, or more probably by their gravitating 
effects upon luminous stars. Thus if one member of a 
Jouble star were dark, we could readily detect its exist- 
ence, and even estimate its size, position, and motions. 

' System of the Worlds tranalated by Harte, vol ii. p. 335 




by observing those of its visible companion. It was a 
favourite notion of Huygher.s that there may exist stars 
and vast universes so distant that their light has never 
yet had time to reach our eyes ; and we must also bear 
in mind that light may possibly suffer slow extinction 
in space, so that there is more than one way in which 
an absolute limit to the powers of telescopic discovery 
may exist 

There are natural limits again to the power of our 
senses in detecting undulations of various kinds. It is 
commonly said that vibrations of more than 38,000 strokes 
per second are not audible as sound ; and as some eare 
actually do hear sounds of much higher pitch, even two 
octaves higher than what other ears can detect, it is 
exceedingly probable that there are incessant vibnitions 
which we cannot call sound because they are never heard. 
Insects may communicate by such acute sounds, con- 
stituting a language inaudible to us ; and the remarkable 
agreement apparent among bodies of ants or bees might 
thus perhaps be explained. Nay, as Fontenelle long ago 
suggested in his scientific romance, there may exist un- 
limited numbers of senses or modes of perception which 
we can never feel, though Darwin's theory woidd render it 
probable that any useful means of knowledge in an an- 
cestor would be developed and improved in the descendants. 
We might doubtless have been endowed with a sense 
capable of feeling electric phenomena with acuteness, so 
that the positive or negative state of charge of a body 
could be at once estimated. The absence of such a 
sense is probably due to its comparative uselessness. 

Heat undulations are subject to the same considerations. 
It is now apparent that what we call light is the affection 
of the eye by certain vibrations, the less rapid of which 
are invisible and constitute the dark rays of radiant heat, 
in detecting which we must substitute the thermometer 
pr the thermopile for the eye. At the other end of the 
spectnim, again, the ultra-violet rays are invisible, and 
only indirectly brought to our knowledge in the pheno- 
mena of fluorescence or photo-chemical action. There is 
no reason to believe that at either end of the spectrum an 
absolute limit has yet been reached. 

Just as our knowledge of the stellar univei-se is limited 



t (^ 













by the power of the telescope and other conditions, so our 
knowledge of the minute world has its limit in the powers 
and optical conditions of the microscope. There was a 
time when it would have been a reasonable induction that 
vegetables are motionless, and animals alone endowed 
with power of locomotion. We are astonished to dis- 
cover by the microscope that minute plants are if any- 
thing more active than minute animals. We even find 
that mineral substances seem to lose their inactive 
character and dance about with incessant motion when 
reduced to suflRciently minute particles, at least when sus- 
pended in a non-conducting medium.^ Microscopists will 
meet a natural limit to observation when the minuteness 
of the objects examined becomes comparable to the length 
of light undulations, and the exti'eme difficulty already 
encountered in determining the forms of minute marks on 
Diatoms appears to be due to this cause. According to 
Helmholtz the smallest distance which can be accurately 
defined depends upon the interference of light passing 
through the centres of the bright spaces. With a the- 
oretically perfect micrascope and a diy lense the smallest 
visible object would not be less than one 8o,ocx)th part 
of an inch in red light. 

Of the errors likely to arise in estimating quantities by 
the senses I have already spoken, but there are some cases 
in which we actually see things diflFerently from what 
they are. A jet of water appears to be a continuous 
thread, when it is really a wonderfully organised succes- 
sion of small and large drops, oscillating in form. The 
drops fall so rapidly that their impressions upon the eye 
i-un into each other, and in order to see the separate drops 
we require some device for giving an instantaneous view. 

One insuperable limit to our powers of observation 
arises from the impossibility of following and identifying 
the ultimate atoms of matter. One atom of oxygen is 
probably undistinguishable from another atom; only by 

^ Tliis curions plicuomcnon, which I propose to call pedetis, or the pedctic 
movanentf from ttjSow, to jump, is carefnlly described in my paper published 
in the Quarterly Journal of Science for April, 1878, vol. riii. (N.S.) 
p. 167. See aI«o Proceedings of the Literary and Philo$cpMcal Society 
tf Manchester, 2Sth Janiuiry, 1870, vol. ix. p. 78, Nature^ 22nd August, 
1878, vol. xTiii. p 44(\ or the QiMrUrly Journal of Science, vol. riii. 
tN.8.)p. 514. 




keeping a certain volume of oxygen safely inclosed in 
a bottle can we assure ourselves of its identity ; allow it 
to mix with other oxygen, and we lose all power of iden- 
tification. Accordingly we seem to have no means of 
directly proving that every gas is in a constant state of 
diffusion of every part into every part. We can only 
infer this to be the case from observing the behaviour 
of distinct gases which we can distinguish in their course, 
and by reasoning on the grounds of molecular theory.^ 

External Conditions of Correct Observation, fy 

Before we proceed to draw inferences from any series of 
recorded facts, we must take care to ascertain perfectly, 
if possible, the external conditions under which the facts 
are brought to our notice. Not only may the observing 
mind be prejudiced and the senses defective, but there 
may be circumstances which cause one kind of event to 
come more frequently to our notice than another. The 
comparative numbers of objects of different kinds existing 
may in any degree differ from the numbers which come to 
our notice. This difference must if possible be taken into 
account before we make any inferences. 

There long appeared to be a strong presumption that 
all comets moved in elliptic orbits, because no comet had 
been proved to move in any other kind of path. The 
theory of gravitation admitted of the existence of comets 
moving in hyperbolic orbits, and the question arose 
whether they were really non-existent or were only 
beyond the bounds of easy observation. From reason- 
able suppositions Laplace calculated that the probability 
was at least 6000 to i against a comet which comes 
within the planetary system sufficiently to be visible at 
the earth's surface, presenting an orbit which could be 
discriminated from a very elongated ellipse or parabola in 
the i>art of its orbit within the reach of our telescopes.^ 
In short, the chances are very much in favour of our 
seeing elliptic rather than hyperbolic comets. Laplace's 
views have been confirmed by the discovery of six 

* Maxwell, Theory of Heat, p. 301. 

« Ijaplace, EsMoi FhilosopkiquCf p. 59. Todhunter's Uiitory, 

pp. 491—494. 












1 1 




hyperbolic comets, which appeared in the years 1729, 
1 77 1, 1774, 18 18, 1840, and 1843,^ and as only about 800 
comets altogether have been recorded, the proportion of 
hyperbolic ones is quite as large as should be expected. 

When we attempt to estimate the numbers of objects 
which may have existed, we must make large allowances 
for the limited sphere of our observations. Probably not 
more than 4000 or 5000 comets have been seen in 
historical times, but making allowance for the absence of 
observers in the southern hemisphere, and for the small 
probability that we see any considerable fraction of those 
which are in the neighbourhood of our system, we must 
^ accept Kepler's opinion, that there are more comets in 
the regions of space than fishes in the depths of the ocean. 
When like calculations are made concerning the numbers 
of meteors visible to us, it is astonishing to find that the 
number of meteors entering the earth's atmosphere in every 
twenty-four hours is probably not less than 400,000,000, 
of which 13,000 exist in every portion of space equal to 
that filled by the earth. 

Serious fsdlacies may arise from overlooking the inevit- 
able conditions under which the records of past events arc 
brought to our notice. Thus it is only the durable objects 
manufactured by former races of men, such as flint imple- 
ments, which can have come to our notice as a general 
rule. The comparative abundance of iron and bronze 
articles used by an ancient nation must not be supposed 
to be coincident with their comparative abundance in our 
museums, because bronze is far Uie more durable. There 
is a prevailing fallacy that our ancestors built more 
strongly than we do, arising from the fact that the more 
fragile structures have long since crumbled away. We 
have few or no relics of the habitations of the poorer 
classes among the Greeks or Eomaus, or in fact of any 
past race ; for the temples, tombs, public buildings, and 
mansions of the wealthier classes alone endure. There is 
an immense expanse of past events necessaiily lost to us 
for ever, and we must generaUy look upon records or relics 
as exceptional in their character. 

The same gonsiderations apply to geological relics. 
We could not generally expect that animals would be 
^ Ciuuubexs' Attronomfff ut ed. |^ 203. 





preserved unless as regaixis the bone^, shells, strong integu- 
ments, or other hard and durable parts. All the infusoria 
and animals devoid of mineral framework have probably 
perished entirely, distilled perhaps into oils. It has been 
pointed out tliat the peculiar character of some extinct 
floras may be due to the unequal preservation of different 
families of plants. By various accidents, however, we gain 
glimpses of a world that is usually lost to us — as by 
insects embedded in amber, the great mammoth preserved 
in ice, mummies, casts in solid material like that of the 
Koman soldier at Pompeii, and so forth. 

We should also remember, that just as there may be 
conjunctions of the heavenly bodies that can have hap- 
pened only once or twice in the period of history, so re- 
markable terrestrial conjunctions may take place. Great 
storms, earthquakes, volcanic eruptions, landslips, floods, 
irruptions of the sea, may, or rather must, have occurred, 
events of such unusual magnitude and such extreme rarity 
that we can neither expect to witness them nor readily 
to comprehend their effects. It is a great advantage of 
the study of probabilities, as Laplace himself remarked, to 
make us mistrust the extent of our knowledge, and pay 
proper regard to the probability that events would come 
within the sphere of our observations. 

Appareiit Sequence of Events, 

De Morgan has excellently pointed out^ that there 
are no less than four modes in which one event may 
seem to follow or be connected with another, without 
being really so. These involve mental, sensual, and ex- 
ternal causes of error, and I will briefly state and illustrate 

Instead of A causing B, it may be our perception of A 
that causes B, Thus it is that prophecies, presentiments, 
and the devices of sorcery and witchcraft often work their 
own ends. A man dies on the day which he has always 
regaixled as his last, from his own fears of the day. An 
incantation effects its purpose, because care is taken to 
frighten the intended victim, by letting him know his 
fate. In all such cases the mental condition is the caus^ 
of apparent coincidence. 

* JSffay on Probabilitiet, Cabinet Cyclopaedia, p. 121. 



[ / 







In a second class of cases, the event A may make our 
perception of B follow, which waidd otherwise happen 
withmU being perceived. Thus it was believed to be the 
result of investigation that more comets appeared in hot 
than cold summers. No account was taken of the fact 
that hot summers would be comparatively cloudless, and 
afford better opportunities for the discovery of comets. 
Here the disturbing condition is of a purely external 
character. Certain ancient philosophers held that the 
moon's rays were cold-producing, mistaking the cold 
caused by radiation into space for an effect of the moon, 
which is more likely to be visible at a time when the 
absence of clouds permits radiation to proceed. 

In a third class of cases, our perception of A may makt 
our perception of B follow. The event B may be con- 
stantly happening, but our attention may not be drawn to 
it except by our observing A. This case seems to be 
illustrated by the fallacy of the moon's influence on clouds. 
The origin of this fallacy is somewhat complicated. In 
the first place, when the sky is densely clouded the moon 
would not be visible at all ; it would be necessary for us to 
see the full moon in order that our attention should be 
strongly drawn to the fact, and this would happen most 
often on those nights when the sky is cloudless. Mr. 
W. Ellis,^ moreover, has ingeniously pointed out that there 
is a general tendency for clouds to disperse at the com- 
mencement of night, which is the time when the full moon 
rises. Thus the change of the sky and the rise of the full 
moon are likely to attract attention mutually, and the 
coincidence in time suggests the relation of cause and 
effect. Mr. Ellis proves from the results of observations 
at the Greenwich Observatory that the moon possesses no 
appreciable power of the kind supposed, and yet it is 
remarkable that so sound an observer as Sir John Herschel 
was convinced of the connection. In his " Results of 
Observations at the Cape of Good Hope,"* he mentions 
many evenings when a full moon occurred with a 
pecidiarly clear sky. 

> Philoiophical Magazine^ 4th Series (1867), toI. xxxiv. p. 64. 

* See NoUi to Measures of Double StarSy 1204, 1336, 1477, 1686, 
1786, 1816, 1835, 1929, 2081, 2186, pp. 265, &c. See also Herschers 
Familiar Lectures on Scientific SubjectSf p 147, and Outlines of 
Astronomy f 7th ed. p. 28c 




V^ There is yet a fourth class of cases, in which B is really 
the arUecedevt event, but our perception of A, which is a 
consequence of B, may he necessary to bring about our 
perception of B. There can be no doubt, for ^instance, 
that upward and downward currents are continually cir- 
culating in the lowest stratum of the atmosphere during 
the day-time ; but owing to the transparency of the at- 
mosphere we have no evidence of their existence until we 
perceive cumulous clouds, which are the consequence of 
such currents. In like manner an interfiltration of bodies 
of air in the higher parts of the atmosphere is probably in 
nearly constant progress, but unless threads of cirrous 
cloud indicate these motions we remain ignorant of their 
occurrence.' The highest strata of the atmosphere are 
wholly imperceptible to us, except when rendered luminous 
by auroral currents of electricity, or by the passage of 
meteoric stones. Most of the visible phenomena of comets 
probably arise from some substance which, existing pre- 
viously invisible, becomes condensed or electrified suddenly 
into a visible form. Sir John Herschel attempted ti) 
explain the production of comet tails in this manner by 
evaporation and condensation.* 

Negative Arguments from Non-ohservation. 

Fiom what has been suggested in preceding sections, it 
will plainly appear that the non-observation of a pheno- 
menon is not generally to be taken as proving its non- 
occurrence. As there are sounds which we cannot hear, 
rays of heat which we cannot feel, multitudes of worlds 
which we cannot see, and myriads of minute organisms 
of which not the most powerful microscope can give us 
a view, we must as a general rule interpret our experience 
in an affirmative sense only. Accordingly when inferences 
have been drawn from the non-occurrence of particular 
facts or objects, more extended and careful examination 
has often proved their falsity. Not many years since it 
was quite a well credited conclusion in geology that no 
remains of man were found in connection with those of 

* Jevons, On the Cxrrous Form of Cloud, Philosophical Magazine, 
July, 1857, 4th Series, vol- xiv. p. 22. 
' Astronomy, 4th ed. p. 358 












extinct animals, or in any deposit not actually at present 
in course of formation. Even Babbage accepted this con- 
clusion as strongly confirmatory of the Mosaic accounts.' 
While the opinion was yet universally held, flint imple- 
ments had been found disproving such a conclusion, and 
overwhelming evidence of man's long-continued existence 
has since been forthcoming. At the end of the last century, 
when Herschel had searched the heavens with his powerful 
telescopes, there seemed little probability that planets yet 
remained unseen within the orbit of Jupiter. But on the 
first day of this century such an opinion was overturned 
by the discovery of Ceres, and more than a hundred other 
small planets have since been added to the lists of the 
planetary system. 

The discovery of the Eozoon Canadense in strata of 
much greater age than any previously known to contain 
organic remains, has given a shock to groundless opinions 
concerning the origin of organic forms; and the oceanic 
dredging expeditions under l3r. Carpenter and Sir Wy ville 
Thomson have modified some opinions of geologists by 
disclosing the continued existence of forms long supposed 
to be extinct. These and many other cases which might 
be quoted show the extremely unsafe character of negative 

But it must not be supposed that negative arguments 
are of no force and value. The earth's surface has been 
sufficiently searched to render it highly improbable that 
any terrestrial animals of the size of a camel remain to be 
discovered. It is believed that no new large animal has 
been encountered in the last eighteen or twenty centuries,* 
and the probability that if existent they would have been 
seen, increases the probability that they do not exist. 
We may with somewhat less confidence discredit the 
existence of any large unrecognised fish, or sed animals, 
such as the alleged sea-serpent. But, as we descend to 
forms of smaller size negative evidence loses weight from 
the less probability of our seeing smaller objects. Even 
the strong induction in favour of the four-fold division of 
the animal kingdom into Vertebrata, Annulosa, Mollusca, 

' Babbage, Ninth Bridgetoater Treatise^ p. 67. 

■ Cuvier, Esiay on the Theory 0/ Uu Earth, trauaUition, p. 61, &C. 

and Ccelenterata, may break down by the discovery of in- 
termediate or anomalous forms. As civilisation spreads 
over the surface of the earth, and unexplored tracts are 
gradually diminished, negative conclusions will increase 
in force ; but we have much to learn yet concerning the 
depths of the ocean, almost wholly unexamined as they 
are, and covering three-fourths of the earth's surface. 

In geology there are many statements to which con- 
siderable probability attaches on account of the large 
extent of the investigations already made, as, for instance, 
that true coal is found only in rocks of a particular geolo- 
gical epoch ; that gold occurs in secondary and tertiary 
strata only in exceedingly small quantities,^ probably 
derived from the disintegration of earlier rocks. In 
natural history negative conclusions are exceedingly 
treacherous and unsatisfactoiy. The utmost patience 
will not enable a microscopist or the observer of any 
living thing to watch the behaviour of the organism under 
all ciVcumstances continuously for a great length of time. 
There is always a chance therefore that the critical act or 
change may take place when the observer's eyes are with- 
drawn. This certainly happens in some cases ; for though 
the fertilisation of orchids by agency of insects is proved 
as well as any fact in natural history, Mr. Darwin has 
never been able by the closest watching to detect an insect 
in the performance of the operation. Mr. Darwin has 
himself adopted one conclusion on negative evidence, 
namely, that the Orchis pyramidalis and certain other 
orchidaceous flowers secrete no nectar. But his caution 
and unwearying patience in verifying the conclusion give 
an impressive lesson to the observer. For twenty-three 
consecutive days, as he tells us, he examined flowers in all 
states of the weather, at all hours, in various localities. 
As the secretion in other flowers sometimes takes place 
rapidly and might happen at early dawn, that inconvenient 
hour of observation was specially adopted. Flowers of 
different ages were subjected to irritating vapours, to mois- 
ture, and to every condition likely to bring on the secretion ; 
and only after invariable failure of this exhaustive inquir}' 
was the barrenness of the nectaries assumed to be proved' 

1 Murchison's SUuriaf ist ed. p. 432. 

* Darwin's Fertiluation of Orehids, p. 48. 














In order that a negative argument founded on the non- 
observation of an object shall have any considerable force, 
it must be shown to be probable that the object if existent 
/<^ would have been observed, and it is this probability which 
defines the value of the negative conclusion. The failure 
of astronomers to see the planet Vulcan, supposed by some 
to exist within Mercury's orbit, is no sufficient disproof of 
its existence. Similarly it would be very difficult, or even 
impossible, to disprove the existence of a second satellite of 
small size revolving round the earth. But if any person ' 
make a particular assertion, assigning place and time, then 
oDservation will either prove or disprove the alleged fiict. 
If it is true that when a French observer professed to 
have seen a planet on the sun's face, an observer in Brazil 
was carefully scrutinising the sun and failed to see it, we 
have a negative proof. False facts in science, it has been 
well said, are more mischievous than false theories. A 
false theory is open to every person's criticism, and is ever 
liable to be judged by its accordance with facts. But a 
false or grossly erroneous assertion of a fact often stands 
in the way of science for a long time, because it may be 
extremely difficult or even impossible to prove the falsity 
of what has been once recorded. 

In other sciences the force of a negative argument will 
often depend upon the number of possible alternatives 
which may exist. It was long believed that the quality 
of a musical sound as distinguished from its piteh, must 
depend upon the form of the undulation, because no other 
cause of it had ever been suggested or was apparently 
possible. The truth of the conclusion was proved by 
Helmholtz, who applied a microscope to luminous points 
attached to the strings of various instruments, and 
thus actually observed the different modes of undulation. 
In mathematics negative inductive arguments have 
seldom much force, because the possible forms of expres- 
sion, or the possible combinations of lints and circles in 
geometry, are quite unlimited in number. An enormous / 
number of attempts were made to trisect the angle by the/ 
ordinary methods of Euclid's geometry, but their in/^ 
variable failure did not establish the impossibility of the 
task. This was shown in a totally different manner, by 
proving that the problem involves an inedudble cubic 





equation to which there could be 
geometrical solution.^ This is 
ahsurdum, a form of argument 
character. Similarly no number 
general solution of equations of 
establish the impossibility of the 
mode, equivalent to a reductio ad 
bility is considered to be proved. 

no corresponding plane 
a case of redicctio ad 
of a totally different 
of failures to obtain a 
the fifth degree would 
t^k, but in an indirect 
absurdum, the impossi- 

^ Peacock, AlgAre, voL ii. p. 344. 

* Ibid, p. 359. Serret, Alg^bre SupSriewt^ and ed. p. 304.. 


s . / 




We may now consider the great advantages which we 
enjoy in examining the combinations of phenomena when 
things are within our reach and capable of being experi- 
mented on. We are said to experiment when we bring sub- 
stances together under various conditions of temperature, 
^ pressure, electric disturbance, chemical action, &c., and 
then record the changes observed. Our object in induc- 
tive investigation is to ascertain exactly the group of cir- 
cumstances or conditions which being present, a certain 
other group of phenomena will follow. If we denote by 
A the antecedent group, and by X subsequent pheno- 
mena, our object will usually be to discover a law of the 
form A = AX, the meaning of which is that where A is X 
will happen. 

The circumstances which might be enumerated as present 
in the simplest experiment are very numerous, in fact al- 
most infinite. Rub two sticks together and consider what 
would be an exhaustive statement of the conditions. 
There are the form, hardness, organic sti-ucture, and all 
the chemical qualities of the wood; the pressure and 
velocity of the rubbing ; the temperature, pressure, and all 
the chemical qualities of the surrounding air ; the proxi- 
mity of the earth with its attractive and electric powers ; 
the temperature and other properties of the persons pro- 
ducing motion ; the radiation from the sun, and to and 
from the sky ; the electric excitement possibly existing in 
any overhanging cloud ; even the positions of the heavenly 
bodies must be mentioned. On d priori grounds it is 




unsafe to assume that any one of these circumstances is 
without effect, and it is only by experience that we can 
single out those precise conditions from which the observed 

^^eat of friction proceeds. 

f J The great method of experiment consists in removing, 

^one at a time, each of those conditions which may be 
imagined to have an influence on the result. Our object 
in the experiment of rubbing sticks is to discover the exact 
circumstances under which heat appears. Kow the pre- 
sence of air may be requisite ; therefore prepare a vacuum, 
and rub the sticks in every respect as before, except that 
it is done in vacuo. If heat still appears we may say that 
air is not, in the presence of the other circumstances, a 
requisite condition. The conduction of heat from neigh- 
bouring bodies may be a condition. Prevent this by mak- 
ing all the surrounding bodies ice cold, which is what Davy 
aimed at in rubbing two pieces of ice together. If heat 
still appears we have eliminated another condition, and so 
we may go on until it becomes apparent that the expen- 
diture of energy in the friction of two bodies is the sole 
condition of the production of heat. 

The great difficulty of experiment arises from the fact 

•// that we must not assume the conditions to be independent 
P revious to experime nt we have no right to say that the 
rubbing of two sticks will produce heat in the same way 
when air is absent as before. We may have heat produced 
in one way when air is present, and in another when air 
is absent The inquiry branches out into two lines, and 
we ought to try in both cases whether cutting off a supply 
of heat by conduction prevents its evolution in friction. 
The same branching out of the inquiry occurs with regard 
to every circumstance which enters into the experiment 
Regarding only four circumstances, say A, B, C, D, we 
ought to test not only the combinations ABCD, ABCrf, 
ABcD, A6CD, aBCD, but we ought really to go through 
the whole of the combinations given in the fifth column 
of the Logical Alphabet. The effect of the absence of 
each condition should be tried both in the presence and 
absence of every other condition, and every selection of 
those conditions. Perfect and exhaustive experimentation 
would, in short, consist in examining natural phenomena 
in all their possible combinations and registering al] 

K E 







relations between conditions and results which are found 
capable of existence. It would thus resemble the exclusion 
of contradictory combinations carried out in the Indirect 
Method of Inference, except that the exclusion of com- 
binations is grounded not on prior logical premises, but 
on a posteriori results of actual trial. 

The reader will perceive, however, that such exhaustive 
investigation is practically impossible, because the number 
of requisite experiments would be immensely great Four 
antecedents only would require sixteen experiments; twelve 
antecedents would require 4096, and the number increases 
as the powers of two. The result is that the experimenter 
lias to fall back upon his own tact and experience in select- 
ing those experiments which are most likely to yield him 
significant facts. It is at this point that logical rules and 
forms begin to fail in giving aid. The logical rule is— Try 
all possible combinations; but this being impmcticable, 
the experimentalist necessarily abandons strict logical 
method, and trusts to his own insight. Analogy, as we 
shall see, gives some assistance, and attention sliould be 
concentrated on those kinds of conditions which have been 
found important in like cases. But we are now entirely 
in the region of probability, and the experimenter, while 
he is confidently pursuing what he thinks the right clue, 
may be overlooking the one condition of importance. It is 
an impressive lesson, for instance, that Newton pursued 
all his exquisite researches on the spectrum unsuspicious of 
the fact that if he reduced the hole in the shutter to a 
narrow slit, all the mysteries of the bright and dark lines 
were within his grasp, provided of course that his prisms 
were sufficiently good to define the rays. In like manner 
we know not what slight alteration in the most familiar 
experiments may not open the way to realms of new 

Practical difficulties, also, encumber the progi-ess of the 
physicist. It is often impossible to alter one condition 
without altering others at the same time; and thus we 
may not get the pure effect of the condition in question. 
Some conditions may be absolutely incapable of alteration ; 
others may be with great difficulty, or only in a certain 
degree, removable. A very treacherous source of error is 
the existence of unknown conditions, which of coujse we 




cannot remove except by accident These difficulties we 
will shortly consider in succession. 

It is beautiful to observe how the alteration of a single 
circumstance sometimes conclusively explains a pheno- 
menon. An instance is found in Faraday's investigation 
of the behaviour of Lycopodium spores scattered on a 
vibrating plate. It was observed that these minute spores 
collected together at the points of greatest motion, whereas 
sand and all heavy particles collected at the nodes, where 
the motion was least It happily occurred to Faraday to 
try the experiment in the exhausted receiver of an air- 
pump, and it was then found that the light powder behaved 
exactly like heavy powder. A conclusive proof was thus 
obtained that the presence of air was the condition of im- 
portance, doubtless because it was thrown into eddies by 
the motion of the plate, and carried the Lycopodium to 
the points of greatest agitation. Sand was too heavy to be 
carried by the air. 

Exclusion of Indifferent Circumstances, 

From what has been already said it will be apparent 
that the detection and exclusion of indifferent circum- 
stances is a work of importance, because it allows the 
concentration of attention upon circumstances which con- 
tain the principal condition. Many beautiful instances may 
be given where all the most obvious antecedents have been 
shown to have no part in the production of a phenomenon. 
A pei-son might suppose that the peculiar colours of mother- 
of-pearl were due to the chemical qualities of the substance 
Much trouble might have been spent in following out that 
notion by comparing the chemical qualities of various iri- 
descent substances. But Brewster accidentally took an; 
impression from a piece of mother-of-pearl in a cement of 
resin and bees'-wax, and finding the colours repeated upon 
the surface of the wax, he proceeded to take other impres- 
sions in balsam, fusible metal, lead, gum arable, isinglass, 
&c., and always found the iridescent colours the same. He 
thus proved that the chemical nature of the substance is a 
matter of indifference, and that the form of the surface is 
the real condition of such colours.^ Nearly the same may 
" frtatiu on, Optiu^ by Brewster, Cab. Cyclo. p. 1 17. 

? F 2 










be said of the colours exhibited by thin plates and films. 
The rings and lines of colour will be nearly the same in 
character whatever may be the nature of the substance ; 
nay, a void space, such as a crack in glass, would produce 
them even though the air were withdrawn by an air-pump. 
The conditions are simply the existence of two reflecting 
surfaces separated by a very small space, though it should 
be added that the refractive index of the intervening sub- 
stance has some influence on the exact nature of the colour 

When a ray of light passes close to the edge of an opaque 
body, a portion of tlie light appears to be bent towards it, 
and produces coloured fringes within the shadow of the 
body. Newton attributed this inflexion of light to the 
attraction of the opaque body for the supposed particles of 
liglit, although he was aware that the nature of the sur- 
rounding medium, whether air or other pellucid substance, 
exercised no apparent influence on the phenomena. 
Gravesande proved, however, that the character of the 
fringes is exactly the same, whether the body be dense or 
rare, compound or elementary. A wire produces exactly 
the same fringes as a hair of the same thickness. Even the 
form of the obstructing edge was subsequently shown to 
be a matter of indifference by Fresnel, and the interfer- 
ence spectrum, or the spectrum seen when light passes 
through a fine grating, is absolutely the same whatever be 
the form or chemical nature of the bars making the 
grating. Thus it appeara that the stoppage of a portion of 
a beam of light is the sole necessary condition for the 
diffraction or inflexion of light, and the phenomenon is 
shown to bear no analogy to the refraction of light, in 
which the form and nature of the substance are all impor- 

It is interesting to observe how carefully Newton, in his 
researches on the spectrum, ascertained the indifference 
of many circumstances by actual trial. He says : * " Now 
the different magnitude of the hole in the window-shut, 
and different thickness of the prism where the rays passed 
through it, and different inclinations of the prism to the 
horizon, made no sensible changes in the length of the 

■ Oflidu, 3rd. ed. p. 2$. 

I '1 





image. Neither did the different matter of the prisms 
make any : for in a vessel made of polished plates of glass 
cemented together in the shape of a prism, and filled with 
water, there is the like success of the experiment according 
to the quantity of the refraction." But in the latter state- 
ment, as I shall afterwards remark (p. 432), Newton 
assumed an indifference which does not exist, and fell 
into an unfortunate mistake. 

In the science of sound it is shown that the pitch of a 
sound depends solely upon the number of impulses in a 
second, and the material exciting those impulses is a matter 
of indifference. Whatever fluid, air or water, gas or liquid, 
be forced into the Siren, the sound produced is the same ; 
and the material of which an organ-pipe is constructed 
does not at all affect the pitch of its sound. In the science 
of statical electricity it is an important principle that the 
nature of the interior of a conducting body is a matter of 
no importance. The electrical charge is confined to the 
conducting surface, and the interior remains in a neutral 
state. A hollow copper sphere takes exactly the same 
charge as a solid sphere of the same metal. 

Some of Faraday's most elegant and successful researches 
were devoted to the exclusion of conditions which previous 
experimentei-s had thought essential for the production of 
electrical phenomena. Davy asserted that no known fluids, 
except such as contain water, could be made the medium 
of connexion between the poles of a battery ; and some 
chemists believed that water was an essential agent in 
electro-chemical decomposition. Faraday gave abundant 
experiments to show that other fluids allowed of elec- 
trolysis, and he attributed the erroneous opinion to the very 
general use of water as a solvent, and its presence in most 
natural bodies.* It was, in fact, upon the weakest kind of 
negative evidence that the opinion had been founded. 

Many experimenters attributed peculiar powers to the 
poles of a battery, likening them to magnets, which, by 
their attractive powers, tear apart the elements of a sub- 
stance. By a beautiful series of experiments,* Faraday 
proved conclusively that, on the contrary, the substance of 

* .fixpenmentoZ Researches in Electrieity, vol. i. pp. 133, 134. 

* Ibid, vol L pp. 127, 162, &c. 

t. ! 






the poles is of no importance, being merely the path 
through which the electric force reaches the liquid acted 
upon. Poles of water, charcoal, and many diverse sub- 
stances, even air itself, produced similar results; if the 
chemical nature of the pole entered at all into the question, 
it was as a disturbing agent. 

It is an essential part of the theory of gravitation that 
the proximity of other attracting particles is without effect 
upon the attraction existing between any two molecules. 
Two pound weights weigh as much together as they do 
separately. Every pair of molecules in the world have, as 
it were, a private communication, apart from their rela- 
tions to all other molecules. Another undoubted result of 
experience pointed out by Newton ^ is that the weight of 
a body does not in the least depend upon its form or 
texture. It may be added that the temperature, electric 
condition, pressure, state of motion, chemical qualities, and 
all other circumstances concerning matter, except its mass, 
are indifferent as regards its gravitating power. 

As natural science progresses, physicists gain a kind of 
insight and tact in judging what qualities of a substance 
are likely to be concerned in any class of phenomena. The 
physical astronomer treats matter in one point of view, 
the chemist in another, and the students of physical optics, 
sound, mechanics, electricity, &c., make a fair division of 
the qualities among them. But errors will arise if too 
much confidence be placed in this independence of various 
kinds of phenomena, so that it is desirable from time to 
time, especially when any unexplained discrepancies come 
into notice, to question the indifference which is assumed 
to exist, and to test its real existence by appropriate 

SimpUJiccUion of Experiments, 

One of the most requisite precautions in experimentation 
is to vary only one circumstance at a time, and to main- 
tain all other cii'cumstances rigidly unchanged. There are 
two distinct reasons for this rule, the first and most ob- 
vious being that if we vary two conditions at a time, and 

' Frineipiaf bk. iii. Prop. vi. Corollary I 


I IX.] 



find some effect, we cannot tell whether the effect is due 
to one or the other condition, or to both jointly. A second 
reason is that if no effect ensues we cannot safely conclude 
that either of them is indifferent ; for the one may have 
neutralised the effect of the other. In our symbolic logic 
AB -I- Ab was shown to be identical with A (p. 97), so 
that B denotes a circumstance which is indifferently 
present or absent. But if B always go together with 
another antecedent C, we cannot show the same inde- 
pendence, for ABC -I- Abe is not identical with A and 
none of our logical processes enables us to reduce it to A. 

If we want to prove that oxygen is necessary to life, we 
must not put a rabbit into a vessel from which the oxygen 
has been exhausted by a burning candle. We should then 
have not only an absence of oxygen, but an addition of 
carbonic acid, which may have been the destructive agent. 
For a similar reason Lavoisier avoided the use of atmo- 
spheric air in experiments on combustion, because air was 
not a simple substance, and the presence of nitrogen might 
impede or even alter the effect of oxygen. As Lavoisier 
remarks,* " In performing experiments, it is a necessary 
principle, which ought never to be deviated from, that 
they be simplified as much as possible, and that every 
circumstance capable of rendering their results complicated 
be carefully removed." It has also been well said by 
Cuvier ' that the method of physical inquiry consists in 
isolating bodies, reducing them to their utmost simplicity, 
and bringing each of their properties separately into action, 
either mentally or by experiment. 

The electro-magnet has been of the utmost service in 
the investigation of the magnetic properties of matter, by 
allowing of the production or removal of a most powerful 
magnetic force without disturbing any of the other ar- 
rangements of the experiment. Many of Faraday's most 
valuable experiments would have been impossible had it 
been necessary to introduce a heavy permanent magnet, 
which could not be suddenly moved without shaking the 
whole apparatus, disturbing the air, producing currents 
by changes of temperature, &c. The electro-magnet is 

» Layoisier's CJumittry, translated by Kerr, p. 103. 
• Cuvier's Animal Kingdom^ introduction. Dp i. 2. 









perfectly under control, and its influence can be brought 
into action, reversed, or stopped by merely touching a 
button. Thus Faraday was enabled to prove the rotation 
of the plane of circularly polarised light by the fact that 
certain light ceased to be visible when the electric current 
of the magnet was cut off, and re-appeared when the 
current was made. " These phenomena," he says, " could 
be reversed at pleasure, and at any instant of time, and 
upon any occasion, showing a perfect dependence of cause 
and effect." ^ 

It was Newton's omission to obtain the solar spectrum 
under the simplest conditions which prevented him from 
discovering the dark lines. Using a broad beam of light 
which had passed through a round hole or a triangular 
slit, he obtained a brilliant spectrum, but one in which 
many different coloured rays overlapped each other. In 
the recent history of the science of the spectrum, one 
main difficulty has consisted in the mixture of the lines of 
several different substances, which are usually to be found 
in the light of any flame or spark. It is seldom possible 
to obtain the light of any element in a perfectly simple 
manner. Angstrom greatly advanced this branch of science 
by examining the light of the electric spark when formed 
between poles of various metals, and in the presence of 
various gases. By varying the pole alone, or the gaseous 
medium alone, he was able to discriminate correctly be- 
tween the lines due to the metal and those due to the 
surrounding gas.^ 

Failure in the Simplification of Experiments, 

In some cases it seems to be impossible to carry out the 
rule of varying one circumstance at a time. When we 
attempt to obtain two instances or two forms of experi- 
ment in which a single circumstance shall be present in 
one case and absent in another, it may be found that this 
single circumstance entails others. Benjamin Franklin's 
experiment concerning the comparative absorbing powers 
of different colours is well known. " I took/* he says, " a 

' Experimental Researches in Electricityf vol. iiL p. 4. 
* Philosophical Magazine, 4th Series, vol. ix. p. 327. 

number of little square pieces of broadcloth from a tailor's 
pattern card, of various colours. They were black, deep 
blue, lighter blue, green, purple, red, yellow, white, and 
other colours and shades of colour. I laid them all out 
upon the snow on a bright sunshiny morning. In a few 
hours the black, being most warmed by the sun, was sunk 
80 low as to be below the stroke of the sun's rays ; the 
dark blue was almost as low ; the lighter blue not quite 
so much as the dark ; the other colours less as they were 
lighter. The white remained on the surface of the snow, 
not having entered it at all." This is a very elegant and 
apparently simple experiment ; but when Leslie had com- 
pleted liis series of researches upon the nature of heat, he 
came to the conclusion that the colour of a surface has 
very little effect upon the radiating power, the mechanical 
nature of the surface appearing to be more influentiaL 
He remarks ^ that " the question is incapable of being posi- 
tively resolved, since no substance can be made to assume 
difTerent colours without at the same time changing its 
internal structure." Recent investigation has shown that 
the subject is one of considerable complication, because 
the absorptive power of a surface may be different accord- 
ing to the character of the rays which fall upon it ; 
but there can be no doubt as to the acuteness with which 
Leslie points out the difficulty. In Well's investigations 
concerning the nature of dew, we have, again, very 
complicated conditions. If we expose plates of various 
material, such as rough iron, glass, polished metal, to the 
midnight sky, they will be dewed in various degrees j 
but since these plates differ both in the nature of the 
surface and the conducting power of the material, it would 
not be plain whether one or both circumstances were of 
importance. We avoid this difficulty by exposing the 
same material polished or varnished, so as to present dif- 
ferent conditions of surface ; * and again by exposing 
different substances with the same kind of surface. 

When we are quite unable to isolate circumstances we 
must resort to the procedure described by Mill under the 
name of the Joint Method of Agreement and Difference 

* Inquiry into tJie Naiure of Heai, p. 95. 

* Herschel, Preliminary Dtscoursey p. 161. 











' I 




We must collect as many instances as possible in which 
a given circumstance produces a given result, and as many 
as possible in which the absence of the circumstance is 
/ followed by the absence of the result. To adduce his 
^ example, we cannot experiment upon the cause of double 
refraction in Iceland spar, because we cannot alter its 
irystalline condition without altering it altogether, nor can 
we find substances exactly like calc spar in every circum- 
stance except one. We resort therefore to the method of 
comparing together all known substances which have the 
property of doubly-refracting light, and we find that they 
agree in being crystalline.^ This indeed is nothing but an 
ordinary process of perfect or probable induction, already 
partially described, and to be further discussed under 
Classification. It may be added that the subject does 
admit of perfect experimental treatment, since glass, when 
compressed in one direction, becomes capable of doubly- 
refracting light, and as there is probably no alteration in 
the glass but change of elasticity, we learn that the power 
of double refraction is probably due to a difference of 
elasticity in different directions. 

BenumU of UsiieU Conditions, 

One of the great objects of experiment is to enable us 
to judge of the behaviour of substances under conditions 
widely different from those which prevail upon the surface 
of the earth. We live in an atmosphere which does not 
vary beyond certain narrow limite in temperature or 
pressure. Many of the powers of nature, such as gravity, 
which constantly act upon us, are of almost fixed amount. 
Now it will afterwards be shown that we cannot apply a 
quantitative law to circumstances nmch differing from 
those in which it was observed. In the other planets, the 
sun, the stars, or remote parts of the Universe, the con- 
ditions of existence must often be widely different from 
what we commonly experience hera Hence our know- 
ledge of nature must remain restricted and hypothetical, 
unless we can subject substances to unusual conditions by 
suitable experiments. 

» Sydem of Logic, bk. iii. chap. viii. § 4, 5th ed, toI. i. p. 433. 




The electric arc is an invaluable means of exposing 
metals or other conducting substances to the highest 
known temperatui*e. By its aid we learn not only that 
all the metals can be vaporised, but that they all give off 
distinctive i-ays of light At the other extremity of the 
scale, the intensely powerful freezing mixture devised by 
Faraday, consisting of solid carbonic acid and ether mixed 
in vacuo, enables us to observe the nature of substances at 
temperatures immensely below any we meet with naturally 
on the earth's surface. 

We can hardly realise now the importance of the in- 
vention of the air-pump, previous to which invention it 
was exceedingly difficult to experiment except under the 
ordinary pressure of the atmosphere. The Torricellian 
vacuum had been employed by the philosophers of the 
Accademia del Cimento to show the behaviour of water, 
smoke, sound, magnets, electric substances, &c., in vacuo, 
but their experiments were often unsuccessful from the 
difficulty of excluding air.^ 

Among the most constaitt circumstances under which 
we live is the force of gravity, which does not vary, except 
by a slight fraction of its amount, in any part of the earth's 
crust or atmosphere to which we can attain. This force is 
sufficient to overbear and disguise various actions, for in- 
stance, the mutual gravitation of small bodies. It was an 
interesting experiment of Plateau to neutralise the action 
of gravity by placing substances in liquids of exactly the 
same specific gravity. Thus a quantity of oil poured into 
the middle of a suitable mixture of alcohol and water 
assumes a spherical shape; on being made to rotate it 
becomes spheroidal, and then successively separates into 
a ring and a group of spherules. Thus we have an 
illustration of the mode in which the, planetary system 
may have been produced,* though the extreme difference 
of scale prevents our arguing with confidence from the 
experiment to the conditions of the nebidar theory. 

It is possible that the so-called elements are elementary 
only to us, because we are restricted to temperatures at 
which they are fixed. Lavoisier carefully defined an 

' Jit$ttyei of Nalwral ^aperimnUs nubde in th$ Accademia del 
Cimento. Englished by Richard Waller, 1684, p. 40, &c 
- Plateau, Taylor's Sc%«iU\/ie Memoirt, toL ir. pp. 16—43. 






Vf I^^B f 



^^H, 1 










element as a substance which cannot be decomposed by 
any knovm means ; but it seems almost certain that some 
series of elements, for instance Iodine, Bromine, and Chlo- 
rine, are really compounds of a simpler substance. We 
must look to the production of intensely high temperatures, 
yet quite beyond our means, for the decomposition of these 
so-called elements. Possibly in this age and part of the 
universe the dissipation of energy has so far proceeded 
that there are no sources of heat sufficiently intense to 
effect the decomposition. 

Interference of Unsuspected Conditions. 

It may happen that we are not aware of all the conditions 
under which our researches are made. Some substance 
^j// may be present or some power may be in action, which 
escapes the most vigilant examination. Not being awai-e 
of its existence, we are unable to take proper measures to 
exclude it, and thus determine the share which it has in 
the results of our experiment^. There can be no doubt 
that the alchemists were misled and encouraged in their 
vain attempts by the unsuspected presence of traces of 
gold and silver in the substances they proposed to trans- 
mute. Lead, as drawn from the smelting furnace, almost 
always contains some silver, and gold is associated with 
many other metals. Thus small quantities of noble metal 
would often appear as the result of experiment and raise 
delusive hopes. 

In more than one case the unsuspected presence of 
common salt in the air has caused great trouble. In 
the early experiments on electrolysis it was found that 
when water was decomposed, an acid and an alkali were 
produced at the poles, together with oxygen and hydrogen. 
In the absence of" any other explanation, some chemists 
rushed to the conclusion that electricity must have the 
power of generating acids and alkalies, and one chemist 
thought he had discovered a new substance called electric 
acid. But Davy proceeded to a systematic investigation 
of the circumstances, by varying the conditions. Changing 
the glass vessel for one of agate or gold, he found that far 
less alkali was produced ; excluding impurities by the use 
of carefully distilled water, he found that the quantities of 






acid and alkali were still further diminished ; and having 
thus obtained a clue to the cause, he completed the ex- 
clusion of impurities by avoiding contact with his fingers, 
and by placing the apparatus under an exhausted receiver, 
no acid or alkali being then detected. It would be difficult 
to meet with a more elegant case of the detection of a 
condition previously unsuspected.^ 

It is remarkable that the presence of common salt in 
the air, proved to exist by Davy, nevertheless continued a 
stumbling-block in the science of spectrum analysis, and 
probably prevented men, such as Brewster, Herschel, and 
Talbot, from anticipating by thirty years the discoveries 
of Bunsen and Kirchhoff. As I pointed out,* the utility 
of the spectrum was known in the middle of the last 
century to Thomas Melvill, a talented Scotch physicist, 
who died at the early age of 27 years.^ But Melvill 
was struck in his examination of coloured flames by the 
extraordinary predominance of homogeneous yellow light, 
which was due to some circumstance escaping his atten- 
tion. Wollaston and Fraunhofer were equally struck by 
the prominence of the yellow line in the spectrum of 
nearly every kind of light. Talbot expressly recommended 
tlie use of the prism for detecting the presence of substances 
by what we now call spectrum analysis, but he found that 
all substances, however different the light they yielded in 
other respects, were identical as regards the production of 
yellow light. Talbot knew that the salts of soda gave this 
coloured light, but in spite of Davy's previous difficulties 
with salt in electrolysis, it did not occur to him to assert 
that where the light is, there sodium must be. He sug- 
gested water as the most likely source of the yellow light, 
because of its frequent presence, but even substances 
which were apparently devoid of water gave the same 
yellow light.* Brewster and Herschel both experimented 

* Philosophical Transactions [1826], vol. cxvi. pp. 388, 389. Works 
of Sir Humphry Davy, vol. v. pp. i — 12. 

* National lieview^ July, 1801, p. 13. 

* His published works are contained in The Edinburgh Physical 
and Literary Essays, vol. ii. p. 34 ; Philosophical Transactions [1753], 
vol xlviii. p. 261 ; see also Morgan's Papers in Philosophical Trans- 
ft^tiiont [1755], vol. IxiT. p. 190. 





Journal of Science, voL v. p. 79. 





♦ . 


upoa flames almost at the same time as Talbot, and 
Herschel unequivocally enounced the principle of spec- 
trum analysis.^ Nevertheless Brewster, after numerous 
experiments attended with great trouble and disappoint- 
ment, found that yellow light might be obtained from the 
combustion of almost any substance. It was not until 
1856 that Swan discovered that an almost infinitesimal 
quantity of sodium chloride, say a millionth part of a grain, 
was sufficient to tinge a flame of a bright yellow colour. 
The universal diffusion of the salts of sodium, joined to 
this unique light-producing power, was thus shown to be 
the unsuspected condition which had destroyed the confi- 
dence of all previous experimenters in the use of the 
prism. Some references concerning the history of this 
curious point are given below.* 

In the science of radiant heat, early inquirers were led 
to the conclusion that radiation proceeded only from the 
surface of a solid, or from a very small depth below it 
But they happened to experiment upon surfaces covered 
by coats of varnish, which is highly athermanous or 
opaque to heat Had they properly varied the character 
of the surface, using a highly diathermanous substance like 
rock salt, they would have obtained very different results.' 

One of the most extraordinary instances of an erroneous 
opinion due to overlooking interfering agents is that con- 
cerning the increase of rainfall near to the earth's surface. 
More than a century ago it was observed that rain-gauges 
placed upon church steeples, house tops, and other elevated 
places, gave considerably less rain than if they were on the 
ground, and it has been recently shown that the variation 
is most rapid in the close neighbourhood of the ground.* 
All kinds of theories have been started to explain this 
phenomenon ; but I have shown ' that it is simply due to 

* Eneyclop<Bdia Mdropolitanaf art LigJU, § 524; Herschcri 
F(Mnxliar Lectures^ p. 266. 

■ Talbot, Philasophienl Magaziney vd Series, vol. ix. p. i (1836); 
Brewster, Transaetiotu of tJu Royal Society 0/ Edinburgh [1823I 
vol. ii. pp.433, 455 ; Swan, ibid. [1856] vol xxL p. 41 1 ; rhilotophical 
Magazine, 4th Series, vol xx. p. 173 [Sept 1860J ; Boecoe, i^^keUntm 
Analytiif Lecture IIL 

* Balfour Stewart, EUmentary Treatiie on Ueai^ p. 192. 

* British Association, Liverpool, 1870. Rewtri on Kaii^faU, p. I7d 

* PhUoiophieal Magaxiue. Dec. 1861. 4th Series, vol. xxii p. 421^ 

i Ii 




the interference of wind, which deflects more or less rain 
from all the gauges which are exposed to it. 

The great magnetic power of iron renders it a source of 
disturbance in magnetic experiments. In building a mag- 
netic observatory great care must therefore be taken that 
no iron is employed in the construction, and that no 
masses of iron are near at hand. In some cases magnetic 
observations have been seriously disturbed by the existence 
of masses of iron ore in the neighbourhood. In Faraday's 
experiments upon feebly magnetic or diamagnetic substances 
he took the greatest precautions against the presence of 
disturbing substances in the copper wire, wax, paper, and 
other articles used in suspending the test objects. It was 
his custom to try the effect of the magnet upon the appa- 
ratus in the absence of the object of experiment, and with- 
out this preliminary trial no confidence could be placed in 
the results.^ Tyndall has also employed the same mode 
for testing the freedom of electro-magnetic coils from iron, 
and was thus enabled to obtain them devoid of any cause 
of disturbance.* It is worthy of notice that in the very 
infancy of the science of magnetism, the acute experimen- 
talist Gilbert correctly accounted for the opinion existing 
in his day that magnets would attract silver, by pointing 
out that the silver contained iron. 

Even when we are not aware by previous experience of 
the probable presence of a special disturbing agent, we 
ought not to assume the absence of unsuspected inter- 
ference. If an experiment is of really high importance, so 
that any considerable branch of science rests upon it, we 
ought to try it again and again, in as varied conditions as 
possibla We should intentionally disturb the apparatus 
in various ways, so as if possible to hit by accident upon 
any weak point Especially when our results are more 
regular than we have fair grounds for anticipating, ought 
we to suspect some peculiarity in the apparatus which 
causes it to measure some other phenomenon than that in 
question, just as Foucault's pendulum almost always in- 
dicates the movement of the axes of its own elliptic path 
instead of the rotation of the globe. 

* Experimental Researches in Electricity , vol. iil p. 84. &c. 

* Lectures on Heal, p. 21. 

5 ! 


' i\ 





It was in this cautious spirit that Baily acted in his 
experiments on the density of the earth. The accuracy 
of his results depended upon the elimination of all disturb- 
ing influences, so that the oscillation of his torsion balance 
should measure gravity alone. Hence he varied the appa- 
ratus in many ways, changing the small balls subject to 
attraction, changing the connecting rod, and the means of 
suspension. He observed the effect of disturbances, such 
as the presence of visitors, the occurrence of violent storms, 
&c., and as no real alteration was produced in the results, 
he confidently attributed them to gravity.* 

Newton would probably have discovered the mode of 
constructing achromatic lenses, but for the unsuspected 
effect of some sugar of lead which he is supposed to have 
dissolved in the water of a prism. He tried, by means of 
a glass prism combined with a water prism, to produce 
dispersion of light without refraction, and if he had 
succeeded there would liave been an obvious mode of 
producing refraction without dispersion. His failure is 
attributed to his adding lead acetate to the water for the 
purpose of increasing its refractive power, the lead having 
a high dispersive power which frustrated his purpose.' 
Judging from Newton's remarks, in the Philosophical 
Transactions, it would appear as if he had not, without 
many unsuccessful trials, despaired of the construction of 
achromatic glasses.' 

The Academicians of Cimento, in their early and in- 
genious experiments upon the vacuum, were often misled 
by the mechanical imperfections of their ap})aratus. They 
concluded that the air had nothing to do with the produc- 
tion of sounds, evidently because their vacuum was not 
sufficiently perfect. Otto von Guericke fell into a like 
mistake in the use of his newly-constructed air-pump, 
doubtless from the unsuspected presence of air sufficiently 
dense to convey the sound of the bell. 

It is hardly requisite to point out that the doctrine of 
spontaneous generation is due to the unsuspected presence 

» Baily, Memoin of tJie Royal Aiironamieal Society, vol. xir. pn. 
29, 30. '' *^*^ 

*- Grant, History of Phytical Aitronomy, p. 531. 

• Fkilotophical Transactiont, abridged by Lowthorp, 4th edition, 
Tol i. p. 20a. 





of germs, even after the most careful efforts to exclude 
them, and in the case of many diseases, both of animals 
and plants, germs which we have no means as yet of de- 
tecting are doubtless the active cause. It has long been 
II subject of dispute, again, whether the plants which spring 
from newly turned land grow from seeds long buried in 
that land, or from seeds brought by the wind. Argument 
is unphilosophical when direct trial can readily be applied ; 
for by turning up some old ground, and covering a portion 
of it with a glass case, the conveyance of seeds by the 
wind can be entirely prevented, and if the same plants 
appear within and without the case, it will become clear 
that the seeds are in the earth. By gross oversight some 
experimenters have thought before now that crops of ry( 
nad sprung up where oats had been sown. 

Blind or Test Experiments, 

Every conclusive experiment necessarily consists in the 
comparison of results between two different combinations 
of circumstances. To give a fair probability that A is the 
cause of X, we must maintain invariable all surrounding 
objects and conditions, and we must then show that where 
A is X is, and where A is not X is not. This cannot really 
be accomplished in a single trial. If, for instance, a 
chemist places a certain suspected substance in Marsh's 
test apparatus, and finds that it gives a small deposit of 
metallic arsenic, he cannot be sure that the arsenic really 
proceeds from the suspected substance ; the impurity of the 
zinc or sulphuric acid may have been the cause"^ of its 
appearance. It is therefore the practice of chemists to 
make what they call a blind experiment, that is to try 
whether arsenic appears in the absence of the suspected 
substance. The same precaution ought to be taken in all 
important analytical operations. Indeed, it is not merely 
a precaution, it is an essential part of any experiment. If 
the blind trial be not made, the chemist merely assumes 
that he knows what would happen. Whenever we assert 
that because A and X are found together A is the cause of 
X. we assume that if A were absent X would be absent. 
But wherever it is possible, we ought not to take this 
as a mere assumption, or even as a matter of inference. 

F ? 










Experience is ultimately the basis of all our inferences, 
but if we can bring immediate experience to bear upon the 
point in question we should not trust to anything more 
remote and liable to error. When Faraday examined the 
magnetic properties ofthe bearing apparatus, in the absence 
of the substance to be experimented on, he really made a 
blind experiment (p. 431). 

We ought, also, to test the accuracy of a method of ex- 
^ periment whenever we can, by introducing known amounts 
^ of the substance or force to be detected. A new analytical 
process for the quantitative estimation of an element 
should be tested by performing it upon a mixture com- 
pounded so as to contain a known quantity of that element. 
The accuracy of the gold assay process greatly depends 
upon the precaution of assaying alloys of gold of exactly 
known composition.^ Gabriel Plattes' works give evidence 
of much scientific spirit, and when discussing the supposed 
merits of the divining rod for the discovery of subterranean 
treasure, he sensibly suggests that the rod should be tried 
in places where veins of metal are known to exist.* 

Negative Results 0/ Experiment. 

When we pay proper regard to the imperfection of all 
measuring instruments and the possible minuteness of 
effects, we shall see much reason for interpreting with 
caution the negative results of experiments. Wo may fail 
to discover the existence of an expected eflFect, not because 
that effect is really non-existent, but because it is of a 
magnitude inappreciable to our senses, or confounded with 
other effects of much greater amount As there is no 
limit on d priori grounds to the smallness of a phenome- 
non, we can never, by a single experiment, prove the 
non-existence of a supjx)sed effect We are always at 
liberty to assume that a certain amount of effect might 
have been detected by greater delicacy of measurement. 
We cannot safely affirm that the moon has no atmosphere 
at all. We may doubtless show that the atmosphere, if 
present, is less dense than the air in the so-called vacuum 

> Jevous in Watta* Dictionary of Chemistry^ vol. iu pp. 936, 937- 
* Discovery of SubUrraneal TretunrA. London, 1639, p. 48. 

of an air-pump, as did Du Sejour. It is equally impossi- 
ble to prove that gravity occupies no time in transmission. 
Laplace indeed ascertained that the velocity of propagation 
of the influence was at least fifty million times greater than 
that of light ; ^ but it does not really follow that it is in- 
stantaneous; and were there any means of detecting the 
action of one star upon another exceedingly distant star, 
we might possibly find an appreciable interval occupied in 
the transmission of the gravitating impulse. Newton 
could not demonstrate the absence of all resistance to 
matter moving through empty space ; but he ascertained by 
an experiment with the pendulum (p. 443), that if such 
resistance existed, it was in amount less than one five- 
thousandth part of the external resistance of the air.* 

A curious instance of false negative inference is fur- 
nished by experiments on light Euler rejected the cor- 
puscular theory on the ground that particles of matter 
moving with the immense velocity of light would possess 
momentum, of which there was no evidence. Bennet had 
attempted to detect the momentum of light by concentrat- 
ing the rays of the sun upon a delicately balanced body. 
Observing no result, it was considered to be proved that 
light had no momentum. Mr. Crookes, however, having 
suspended thin vanes, blacked on one side, in a nearly 
vacuous globe, found that they move under the influence 
of light. It is now allowed that this effect can be ex- 
plained in accordance with the undulatoiy theory of light, 
and the molecular theory of gases. It comes to this— that 
Bennet failed to detect an effect which he might have 
detected with a better method of experimenting ; but if he 
had found it, the phenomenon would have confirmed, not 
the corpuscular theory of light, as was expected, but the 
rival undulatory theory. The conclusion drawn from 
Bennet's experiment was falsely drawn, but it was never- 
theless true in matter. 

Many incidents in the history of science tend to show 
that phenomena, which one generation has failed to dis- 
cover, may become accurately known to a succeeding 
generation. The compressibility of water which the 

» Uplace, Syttem of the World, translated by Harte, voL ii. p. 322. 

Jruunpta, hk. ii. aect 6, Prop, xxxi Motte's translation. voL u. 
p. 108. , *. ". 

F F 2 




Academicians of Florenco could not detect, because at a 
low pressure the effect was too small to perceive, and at a 
high pressure the water oozed through their silver vessel^ 
has now become the subject of exact measurement and 
precise calculation. Independently of Newton, Hooke 
entertained very remarkable notions concerning the nature 
of gravitation. In this and other subjects he showed, 
indeed, a genius for experimental investigation which 
would have placed him in the first rank in any other age 
than that of Newton. He correctly conceived that the 
force of gravity would decrease as we recede from the 
centre of the earth, and he boldly attempted to prove it by 
experiment Having exactly counterpoised two weights 
in the scales of a balance, or rather one weight against 
another weight and a long piece of fine cord, he removed 
his balance to the top of the dome of St Paul's, and tried 
whether the balance remained in equilibrium after one 
weight was allowed to hang down to a depth of 240 feet. 
No difference could be perceived when the weights were at 
the same and at different levels, but Hooke rightly held 
that the failure arose from the insufficient elevation. He 
says " Yet 1 am apt to think some difference might be dis- 
covered in greater heights." " ITie radius of the earth 
being about 20,922,000 feet, we can now readily calculate 
from the law of gravity that a height of 240 would not 
make a greater difference than one part m 40,000 of the 
weight Such a difference would doubtless be inappreciable 
in the balances of that day, though it could readily be de- 
tected by balances now frequently constructed. Again, the 
mutual gravitation of bodies at the earth's surface is so 
small that Newton appears to have made no attempt to 
demonstrate its existence experimentally, merely remark- 
ing that it was too small to fall under the observation of 
our senses.* It has since been successfully detected and 
measured by Cavendish, Baily, and others. 

The smallness of the quantities which we can sometimes 
observe is astonishing. A balance will weigh to one 
millionth part of the load. Wliitworth can measure to 
the mDlionth part of an inch. A rise of temperature of 

< Euayeg of Natural Erperimenttj &c. p. 1 17. 

• Hookers Posthumous IVorks, p. 1 82. 

* Primcipia, bk. iii Prop. tIl Corollaiy u 




the 88ootli part of a degree centigrade has been detected 
by Dr. Joule. The spectroscope has revealed the presence 
of the 1 0,000,000th part of a gram. It is said that the 
eye can observe the colour produced in a drop of water by 
the 50,000,000th part of a gram of fuschine, and about the 
same quantity of cyanine. By the sense of smell we can 
probably feel still smaller quantities of odorous matter.^ 
We must nevertheless remember that quantitative effects 
of far less amount than these must exist, and we should" 
state our negative results with corresponding caution. We 
can only disprove the existence of a quantitative phenome- 
non by showing deductively from the laws of nature, that 
if present it would amount to a perceptible quantity. As 
in the case of other negative arguments (p. 414), we must 
demonstrate that the effect would appear, where it is b/ 
experiment found not to appear. 

• • 

Limits of Experiment. 

It will be obvious that there are many operations of 
nature which we are quite incapable of imitating in our 
ex|)eriments. Our object is to study the conditions under 
which a certain effect is produced ; but one of tliose con- 
ditions may involve a gieat length of time. There are 
instances on record of experiments extending over five or 
ten years, and even over a large part of a lifetime ; but 
such intervals of time are almost nothing to the time 
during which nature may have l)een at work. The con^ 
tents of a mineral vein in Cornwall may have been under- 
going gradual change for a hundred million years. All 
metamorphic rocks have doubtless endured high tempera- 
ture and enormous pressure for inconceivable periods of 
time, so that chemical geology is generally beyond the 
scope of experiment 

Arguments have been brought against Darwin's theory, 
founded upon the absence of any clear instance of the 
pix)duction of a new species. During an historical interval 
of perhaps four thousand years, no animal, it is said, has 
been so much domesticated as to become different in 




I . 

* KeilPs Introduetion to Natural Philosophy^ 3i-d ed., London, 
*733. pp. 48-54- 




[CH. XIZ. 

species. It might as well be argued that no geological 
changes are taking place, because no new mountain has 
risen in Great Britain within the memory of man. Our 
actual experience of geological changes is like a point in 
the infinite progression of time. When we know that rain 
water falling on limestone will carry away a minute 
portion of the rock in solution, we do not hesitate to 
multiply that quantity by millions, and infer that in 
course of time a mountain may be dissolved away. We 
have actual experience concerning the rise of land in some 
parts of the globe and its fall in others to the extent of 
some feet. Do we hesitate to infer what may thus be done 
in course of geological ages?" As Gabriel Plattos long ago 
remarked, " The sea never resting, but perpetually winning 
land in one place and losing in another, doth show what 
may be done in length of time by a continual operation, 
not subject unto ceasing or intermission." ^ The action of 
physical circumstances upon the forms and characters of 
animals by natural selection is subject to exactly the same 
remarks. As regards animals living in a state of nature, 
the change of circumstances which can be ascertained to 
have occurred is so slight, that we could not expect to 
observe any change in those animals whatever. Nature 
has made no experiment at all for us within historical 
times. Man, however, by taming and domesticating dogs, 
horses, oxen, pigeons, &c., has made considerable change 
in their circumstances, and we find considerable change 
also in their forms and characters. Supposing the state of 
domestication to continue unchanged, these new forms 
would continue permanent so far as we know, and in this 
sense they are permanent. Thus the arguments against 
Darwin's theory, founded on the non-observation of natural 
changes within the historical period, are of the weakest 
character, being purely negative. 

I IHiCcvery of Subkrraneal Treature, 1639, p. 53. 

,: I 



ExFERiMENTS may be of two kinds, experiments of 
simple fact, and experiments of quantity. In the first 
class of experiments we combine certain conditions, and 
wish to ascertain whether or not a certain effect of any 
quantity exists. Hooke wished to ascertain whether or 
not there was any difference in the force of gravity at the 
top and bottom of St. Paul's Cathedral. The chemist 
continually performs analyses for the purpose of ascertaining 
whether or not a given element exists in a particular mi- 
neral or mixture ; all such experiments and analyses are 
qualitative rather than quantitative, because though the 
result may be more or less, the particular amount of the 
result is not the object of the inquiry. 

So soon, however, as a result is known to be discoverable, 
the scientific man ought to proceed to the quantitative p^ 
inquiry, how great a result follows from a certain amounJrQ 
of the conditions which are supposed to constitute ^ 
cause ? The possible numbers of experiments are now in- 
finitely great, for every variation in a quantitative condition 
will usually produce a variation in the amount of the effect. 
The method of variation which thus arises is no narrow or 
special method, but it is the general application of experi- 
ment to phenomena capable of continuous variation. As 
Mr. Fowler has well remarked,^ the observation of variations 
is really an integration of a supposed infinite number of 
applications of the so-called method of difference, that is 
«f axperiment in its perfect foi-m. 

* Elements of Inductive Logic^ ist edit. p. 175. 

■1 i 

1 ; 








Id induction we aim at establishing a general law, and 
if we deal with quantities that law must really be expressed 
more or less obviously in the form of an equation, or 
equations. We treat as before of conditions, ahd of what 
happens under those conditions. But the conditions will 
now vary, not in quality, but quantity, and the effect will 
also vaiy in quantity, so that the result of quantitative in- 
duction is always to arrive at some mathematical expression 
involving the quantity of each condition, and expi-essing 
the quantity of the result. In other words, we wish to 
know what function the effect is of its conditions. We 
shall find that it is one thing to obtain the numerical 
results, and quite another thing to detect the law obeyed 
by those results, the latter being an operation of an inverse 
and tentative character. 


The Variable and the Variant, 

Almost every series of quantitative experiments is 
directed to obtain the relation between the different 
values of one quantity which is varied at will, and au- 
r\ other quantity which is caused thereby to vary. We 
may conveniently distinguish these as respectively the 
variable and the variant. When we are examining the 
effect of heat in expanding bodies, heat, or one of its 
dimensions, temperature, is the variable, length the 
variant. If we compress a body to observe how much 
it is thereby heated, pressure, or it may be the dimensions 
of the body, forms the variable, heat the variant. In 
the thermo-electric pile we make heat the variable and 
measui-e electricity aa the variant. That one of the two 
measui-ed quantities which is an antecedent condition of 
the other will be the variable. 

It is always convenient to have the variable entirely 
under our command. Experiments may indeed be made 
with accuracy, provided we can exactly measure the vari- 
able at the moment when the quantity of the effect is 
determined. But if we have to trust to the action of 
some capricious force, there may be great difficulty in 
making exact measurements, and those results may not 
be disposed over the whole range of quantity in a con- 
venient manner. It is one prime object of the experi- 




menter, therefore, to obtain a regular and governable 
supply of the force which he is investigating. To de- 
tennine correctly the efficiency of windmills, when the 
natural winds were constantly varying in force, would be 
exceedingly difficult Smeaton, therefore, in his experi- 
ments on the subject, created a uniform Avind of the 
required force by moving his models against the air on the 
extremity of a revolving arm.^ The velocity of the wind 
could thus be i*endered greater or less, it could be main- 
tained uniform for any length of time, and its amount 
could be exactly ascertained. In determining the laws of 
the chemical action of light it would be out of the question 
to employ the rays of the sun, which vary in intensity with 
the clearness of the atmosphere, and with every passing 
cloud. One gi-eat difficulty in photometry and the investi- 
gation of the chemical action of light consists in obtaining 
a uniform and governable source of light rays.^ 

Fizeau's method of measuring the velocity of light 
enabled him to appreciate the time occupied by light in 
travelling through a distance of eight or nine thousand 
metres. But the revolving mirror of Wheatstone sub* 
sequently enabled Foucault and Fizeau to measure the 
velocity in a space of four metres. In this latter method 
there was the advantage that various media could be sub- 
stituted for air. and the temperature, density, and other 
conditions of the expeiiment could be accurately governed 
and measured. 

Measurement of the Variable. 

There is little use in obtaining exact measurements of 
ftn effect unless we can also exactly measure its conditions. 

It is absurd to measure the electrical resistance of a 
piece of metal, its elasticity, tenacity, density, or other 
physical qualities, if these vary, not only with the minute 
impurities of the metal, but also with its physical con- 
dition. If the same bar changes its properties by being 

* PhUotophical Transadunu, vol. IL p. 138 ; abridgment, vol. xi 

p. 355. 

> See Bunsen and Roscoe's researches, ia PhUotophical Transaction 
(1859), vol. cxiix. p. 880y ^.f where they describe a constant flame of 
G»rbon iiumoxide eas. 










heated and cooled, and we cannot exactly define the state 
in which it is at any moment, our care in measuring will 
be wasted, because it can lead to no law. It is of little 
use to determine very exactly the electric conductibility of 
carbon, which as graphite or gas carbon conducts like a 
metal, a& diamond is almost a non-conductor, and in 
several other forms possesses variable and intermediate 
powers of conduction. It will bo of use only for 
immediate practical applications. Before measuring these 
we ought to have something to measure of which the con- 
ditions are capable of exact definition, and to which at a 
future time we can recur. Similarly the accuracy of our 
measurement need not much surpass the accuracy with 
which we can define the conditions of the object treated. 

The speed of electricity in passing through a conductor 
mainly depends upon the inductive capacity of the sur- 
rounding substances, and, except for technical or special 
purposes, there is little use in measuring velocities which 
in some cases are one hundred times as great as in other 
cases. But the maxinmm speed of electric conduction is 
probably a constant quantity of great scientific importance, 
and according to Prof. Clerk Maxwell's determination in 
1868 is 174,800 miles per second, or little less than that 
of light. The true boiling point of water is a point on 
which practical thermometry depends, and it is liighly 
important to determine that point in relation to the ab- 
solute thermometric scale. But when water free from air 
and impurity is heated there seems to be no definite limit 
to the temperature it may reach, a temperature of 180** 
Cent, having been actually observed. Such temperatures, 
therefore, do not require accurate measurement. All 
meteorological measuremeuta depending on the accidental 
condition of the sky are of far less importance than 
physical measurements in which such accidental con- 
ditions do not intervene. Many profound investigations 
depend upon our knowledge of the radiant energy con- 
tinually poured upon the earth by the sun ; but this must 
be measured when the sky is perfectly clear, and the 
absorption of the atmosphere at its minimum. The 
slightest interference of cloud destroys the value of such 
a measurement, except for meteorological purposes, whicli 
are of vastly less generality and impoi-tance. it is seldom 




useful, again, to measure the height of a snow-covered 
mountain within a foot, when the thickness of the snow 
alone may cause it to vary 25 feet or more, when in short 
the height itself is indefinite to that extent.^ 

Maintenance of Similar Conditions, 

Our ultimate object in induction must be to obtain the 
complete relation between the conditions and the effect, 
but this relation will generally be so complex that we can 
only attack it in detail We must, as far as possible, 
confine the variation to one condition at a time, and estab- 
lish a separate relation between each condition and the 
effect. This is at any rate the first step in approximating 
to the complete law, and it will be a subsequent question 
how far the simultaneous variation of several conditions 
modifies their separate actions. In many experiments, 
indeed, it is only one condition which we wish to study, 
and the others are interfering forces which we would avoid 
if possible. One of the conditions of the motion of a pen- 
dulum is the resistance of the air, or other medium in 
which it swings ; but when Newton was desirous of prov- 
ing the equal gravitation of all substances, he had no 
interest in the air. His object was to observe a single 
force only, and so it is in a great many other experiments. 
Accordingly, one of the most important precautions in 
investigation consists in maintaining all conditions con- 
stant except that which is to be studied. As that admir- 
able experimental philosopher, Gilbert, expressed it,^ 
" There is always need of similar preparation, of similar 
figure, and of equal magnitude, for in dissimilar and un- 
equal circumstances the experiment is doubtful." 

In Newton's decisive experiment similar conditions were 
provided for, with the simplicity which characterises the 
highest art. The pendulums of which the oscillations were 
compared consisted of equal boxes of wood, hanging by 
equal threads, and filled with different substances, so that 
the total weights should be equal and the centres of oscil- 
lation at the same distance from the points of suspension. 

* Humboldt's Cosmos (Bohn), vol. L p. 7. 
' Giibeit, De Magnete, p. 109. 


i ! 

! i 





Hence the resistauce of the air became approximately a 
matter of indifference ; for the outward size and shape of 
the pendulums being the same, the absolute force of re- 
sistance would be the same, so long as the pendulums 
vibrated with equal velocity ; and the weights being equal 
the resistance would diminish the velocity equally. Hence 
if any inequality were observed in the vibrations of the two 
pendulums, it must arise from the only circumstance which 
was different, namely the chemical nature of the matter 
within the boxes. No inequality being observed, the 
chemical nature of substances can have no appreciable 
influence upon the force of gravitation.* 

A beautiful experiment was devised by Dr. Joule for 
the purpose of showing that the gain or loss of heat by a 
gas is connected, not with the mere change of its volume 
and density, but with the energy received or given out by 
the gaa. Two strong vessels, connected by a tube and stop- 
cock «¥ere placed in water after the air had been exhausted 
from one vessel and condensed in the other to the extent 
of twenty atmospheres. The whole apparatus having 
been brought to a uniform temperature by agitating the 
water, and the temperature having been exactly observed, 
the stopcock was opened, so that the air at once expanded 
and tilled the two vessels uniformly. The temperature of 
the water being again noted was found to be almost un- 
changed. The experiment was then repeated in an exactly 
similar manner, except that the strong vessels were placed 
in separate portions of the water. Now cold was produced 
in the vessel from which the air rushed, and an almost 
exactly equal quantity of heat appeared in that to which 
it was conducted. Thus Dr. Joule clearly proved that 
rarefaction produces as much heat as cold, and that only 
when there is disappearance of mechanical energy will 
there be production of heat.* What we have to notice, 
liowever, is not so much the result of the experiment, as 
the simple manner in which a single change in the appa- 
ratus, the separation of the portions of water surrounding 
the air vessels, is made to give indications of the utmost 

» Pr%neif%€ty bk. iii. Prop. ri. 

'* Phiilosipkical MagiuiiUf 3rd Series, toL xxTi p. 375. 




Collective Experiments. 


There is an interesting class of experiments which 
enable us to observe a number of quantitative results in 
one act. Generally speaking, each experiment yields us 
but one number, and before we can approach the real 
processes of reasoning we must laboriously repeat measure- 
ment after measurement, until we can lay out a curve of 
the variation of one quantity as depending on another. 
We can sometimes abbreviate this labour, by making a 
quantity vary in different parts of the same apparatus 
through every required amount. In observing the height 
to which water rises by the capillary attraction of a glass 
vessel, we may take a series of glass tubes of different 
bore, and measure the height through which it rise^ in each. 
But if we take two glass plates, and place them vertically 
in water, so as to be in contact at one vertical side, and 
slightly separated at the other side, the interval between 
the plates varies through every intermediate width, and 
the water rises to a corresponding height, producing at its 
upper surface a hyperbolic curve. 

The absorption of light in passing through a coloured 
liquid may be beautifully shown by enclosing the liquid in 
a wedge-shaped glass, so that we have at a single glance 
an infinite variety of thicknesses in view. As Newton 
himself remarked, a red liquid viewed in this manner is 
found to have a pale yellow colour at the thinnest part, 
and it passes through orange into red, which gradually 
becomes of a deeper and darker tint.* The effect may be 
noticed in a conical wine-glass. The prismatic analysis of 
light from such a wedge-shaped vessel discloses the reason, 
by exhibiting the progressive absorption of different rays 
of the spectrum as investigated by